12th Maths Monthly Test - 1 ( Applications of Vector Algebra )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 ( Applications of Vector Algebra )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ 3k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) then \(\overset { \rightarrow }{ a } +\left( -\overset { \rightarrow }{ b } \right) \)will be perpendiculur to \(\overset { \rightarrow }{ c } \) only when t =

5
4
3
\(\frac { 7 }{ 3 } \)

The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is
0°

30°

45°

90°
If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec { b } .\vec { c } \) \(\neq \) 0 and \(\vec { a } .\vec { b } \) \(\neq \) 0 then \(\vec { a } \) and \(\vec { c } \) are
perpendicular

parallel

inclined at an angle \(\frac{\pi}{3}\)

inclined at an angle \(\frac{\pi}{6}\)

If the vectors \(a\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \), \(\overset { \wedge }{ i } +b\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +c\overset { \wedge }{ k } \) (a ≠ b ≠ c ≠ 1) are coplaner, then \(\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } =\)

0
1
2
\(\frac { abc }{ (1-a)(1-b)(1-c) } \)

If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \) , then

\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 1
\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= -1
\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 0
\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 2

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

\(\vec { a } \)
\(\vec { b} \)
\(\vec { c } \)
\(\vec { 0 } \)

The length of the 丄^r from the origin to plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 3i } +4\overset { \wedge }{ j } +12\overset { \wedge }{ k } \right) \)= 26 is _____________

2
\(\frac { 1 }{ 2 } \)
26
\(\frac { 26 }{ 169 } \)

The angle between the planes 2x + y - z = 9 and x + 2y + z = 7 is_____________

cos^-1 (5/6)
cos^-1 (5/36)
cos^-1 (1/2)
cos^-1 (1/12)

If \(\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1\)such that \(\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b } \) and \(5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \) are perpendicular to each other, then the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is

45^o
60^o
cos^-1 \(\left( \frac { 1 }{ 3 } \right) \)
cos^-1 \(\left( \frac { 2 }{ 7 } \right) \)

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between

\(\frac { \pi }{ 2 } \)
\(\frac { 3\pi }{ 4 } \)
\(\frac { \pi }{ 4 } \)
\( { \pi }\)

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

\(\frac { \sqrt { 7 } }{ 2\sqrt { 2 } } \)
\(\frac{7}{2}\)
\(\frac { \sqrt { 7 } }{ 2 } \)
\(\frac { 7 }{ 2\sqrt { 2 } } \)

The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { i } -\hat { k) } \) represents a straight line passing through the points

(0,6,1)− and (1,2,1)
(0,6,-1) and (1,4,2)
(1,-2,-1) and (1,4,-2)
(1,-2,-1) and (0,-6,1)

The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =\overset { \wedge }{ 3i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) is _______________

4
\(2\sqrt { 3 } \)
\(4\sqrt { 3 } \)
\(5\sqrt { 3 } \)

If the work done by a force \(\overset { \rightarrow }{ F } =\overset { \wedge }{ i } +m\overset { \wedge }{ j } -\overset { \wedge }{ k } \) in moving the point of application from(1, 1, 1) to (3, 3, 3) along a straight line is 12 units, then m is

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectors, then the vectors \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) \) is parallel to the vector

\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)
\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)
2\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)
2\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)

If \(\overset { \rightarrow }{ p } \times \overset { \rightarrow }{ q } =2\overset { \wedge }{ i } +3\overset { \wedge }{ j } \), \(\overset { \rightarrow }{ r } \times \overset { \rightarrow }{ s } =3\overset { \wedge }{ i } +2\overset { \wedge }{ k } \) then \(\overset { \rightarrow }{ p } .\left( \overset { \rightarrow }{ q } \left( \overset { \rightarrow }{ r } \times \overset { \rightarrow }{ s } \right) \right) \) is

Part - B

Generic 2 - Answer All Question

Forces of magnit \(5\sqrt { 2 } \) and \(5\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } +\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.

A particle acted on by constant forces \(8\hat i+2\hat j-6\hat k\) and \(6\hat i+2\hat j -2\hat k\) is displaced from the point (1, 2, 3) to the point (5, 4, 1). Find the total work done by the forces.

Find the point of intersection of the line x - 1 = \(\frac { y }{ 2 } \) = z + 1 with the plane 2x - y + 2z = 2. Also, find the angle between the line and the plane.

Show that the lines \(\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 } \) and \(\frac { x-3 }{ -2 } =\frac { y-3 }{ 3 } =\frac { 5-z }{ 6 } \) are parallel.

Find the angle between the following lines.
\(\frac { x+4 }{ 3 } =\frac { y-7 }{ 4 } =\frac { z+5 }{ 5 } \), \(\vec { r } =4\hat { k } +t(2\hat { i } +\hat { j } +\hat { k } )\)

Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65 . Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

Let \(\vec { a } ,\vec { b } ,\vec { c } \) be three non-zero vectors such that \(\vec { c } \) is a unit vector perpendicular to both \(\vec { a } \) and \(\vec { b } \). If the angle between \(\vec { a } \) and \(\vec { b } \) is \(\frac { \pi }{ 6 } \), show that \({ [\vec { a } ,\vec { b } ,\vec { c } ] }^{ 2 }\) = \(\frac { 1 }{ 4 } { \left| \vec { a } \right| }^{ 2 }{ \left| \vec { b } \right| }^{ 2 }\)

If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x+1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.

Find the torque of the resultant of the three forces represented by \(-3\hat { i } +6\hat { j } +3\hat { k } \), \(4\hat { i } -10\hat { j } +12\hat { k } \) and \(\hat { 4i } +\hat { 7j } \) acting at the point with position vector \(8\hat { i } -6\hat { j } -4\hat { k } \), about the point with position vector \(18\hat { i } +3\hat { j } -9\hat { k } \)

Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance
between the lines.

If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\)=and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.

If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.
Find the magnitude and direction cosines of the torque of a force represented by \(3\hat { i } +4\hat { j } -5\hat { k } \) about the point with position vector \(2\hat { i } -3\hat { j } +4\hat { k } \) acting through a point whose position vector is \(4\hat { i } +2\hat { j } -3\hat { k } \).

Find the parametric form of vector equation and Cartesian equations of a straight line passing through (5, 2,8) and is perpendicular to the straight lines
\(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )+s(2\hat { i } -2\hat { j } +\hat { k } )\)
\(\vec { r } =(\hat { 2i } -\hat { j } -3\hat { k } )+t(\hat { i } +2\hat { j } +2\hat { k } )\).

Find the parametric form of vector equation, and Cartesian equations of the plane containing the line \(\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )\) and perpendicular to plane \(\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8\)
Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y +4z + 11 = 0, and the point (-2, 1, 3)

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BC } \right| \).

Part - C

Generic 3 - Answer All Question

Find the angle between the lines \(\vec { r } =(\hat { i } +2\hat { j } +4\hat { k } )+t(2\hat { i } +2\hat { j } +\hat { k } )\) and the straight line passing through the points (5, 1, 4) and (9, 2, 12)

If \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) then show that \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } =\overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \)

Find the angle between the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0

Find the equation of the plane passing through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0\) and \(\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2\) and the point (-1, 2, 1)
Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

A straight line passes through the point (1, 2, −3) and parallel to \(4\hat { i } +5\hat { j } -7\hat { k } \). Find
(i) vector equation in parametric form
(ii) vector equation in non-parametric form
(iii) Cartesian equations of the straight line.

Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect
Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + Z + 1 - 0 and perpendicular to the plane x + 2y - 3z + 6 = 0

Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a²=b²+c²−2bc cos A
(ii) b²=c²+a²−2ca cos B
(iii) c²= a²+b²−2ab cos C

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a=bcosC+ccos B
(ii) b=ccosA+acosC
(iii) c=acosB+bcos A

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)

Part - D

Generic 5 - Answer All Question

Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38
ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If. the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)

If the Cartesian equation of a plane is 3x - 4y + 3z = -8, find the vector equation of the plane in the standard form.

If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \)=3

Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)

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12th Maths Monthly Test - 1 ( Applications of Vector Algebra )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 ( Applications of Vector Algebra )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \) , then

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

The length of the 丄r from the origin to plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 3i } +4\overset { \wedge }{ j } +12\overset { \wedge }{ k } \right) \)= 26 is _____________

The angle between the planes 2x + y - z = 9 and x + 2y + z = 7 is_____________

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { i } -\hat { k) } \) represents a straight line passing through the points

The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =\overset { \wedge }{ 3i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) is _______________

If the work done by a force \(\overset { \rightarrow }{ F } =\overset { \wedge }{ i } +m\overset { \wedge }{ j } -\overset { \wedge }{ k } \) in moving the point of application from(1, 1, 1) to (3, 3, 3) along a straight line is 12 units, then m is

Part - B

Generic 2 - Answer All Question

A particle acted on by constant forces \(8\hat i+2\hat j-6\hat k\) and \(6\hat i+2\hat j -2\hat k\) is displaced from the point (1, 2, 3) to the point (5, 4, 1). Find the total work done by the forces.

Find the point of intersection of the line x - 1 = \(\frac { y }{ 2 } \) = z + 1 with the plane 2x - y + 2z = 2. Also, find the angle between the line and the plane.

Show that the lines \(\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 } \) and \(\frac { x-3 }{ -2 } =\frac { y-3 }{ 3 } =\frac { 5-z }{ 6 } \) are parallel.

Find the angle between the following lines. \(\frac { x+4 }{ 3 } =\frac { y-7 }{ 4 } =\frac { z+5 }{ 5 } \), \(\vec { r } =4\hat { k } +t(2\hat { i } +\hat { j } +\hat { k } )\)

Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65 . Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x+1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.

Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance between the lines.

If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.

Find the magnitude and direction cosines of the torque of a force represented by \(3\hat { i } +4\hat { j } -5\hat { k } \) about the point with position vector \(2\hat { i } -3\hat { j } +4\hat { k } \) acting through a point whose position vector is \(4\hat { i } +2\hat { j } -3\hat { k } \).

Find the parametric form of vector equation, and Cartesian equations of the plane containing the line \(\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )\) and perpendicular to plane \(\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8\)

Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y +4z + 11 = 0, and the point (-2, 1, 3)

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BC } \right| \).

Part - C

Generic 3 - Answer All Question

Find the angle between the lines \(\vec { r } =(\hat { i } +2\hat { j } +4\hat { k } )+t(2\hat { i } +2\hat { j } +\hat { k } )\) and the straight line passing through the points (5, 1, 4) and (9, 2, 12)

Find the angle between the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0

Find the equation of the plane passing through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0\) and \(\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2\) and the point (-1, 2, 1)

Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

A straight line passes through the point (1, 2, −3) and parallel to \(4\hat { i } +5\hat { j } -7\hat { k } \). Find (i) vector equation in parametric form (ii) vector equation in non-parametric form (iii) Cartesian equations of the straight line.

Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + Z + 1 - 0 and perpendicular to the plane x + 2y - 3z + 6 = 0

Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin

With usual notations, in any triangle ABC, prove the following by vector method. (i) a2=b2+c2−2bc cos A (ii) b2=c2+a2−2ca cos B (iii) c2= a2+b2−2ab cos C

With usual notations, in any triangle ABC, prove the following by vector method. (i) a=bcosC+ccos B (ii) b=ccosA+acosC (iii) c=acosB+bcos A

Part - D

Generic 5 - Answer All Question

Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38

If the Cartesian equation of a plane is 3x - 4y + 3z = -8, find the vector equation of the plane in the standard form.

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)

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The length of the 丄^r from the origin to plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 3i } +4\overset { \wedge }{ j } +12\overset { \wedge }{ k } \right) \)= 26 is _____________

Find the angle between the following lines.
\(\frac { x+4 }{ 3 } =\frac { y-7 }{ 4 } =\frac { z+5 }{ 5 } \), \(\vec { r } =4\hat { k } +t(2\hat { i } +\hat { j } +\hat { k } )\)

Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance
between the lines.

A straight line passes through the point (1, 2, −3) and parallel to \(4\hat { i } +5\hat { j } -7\hat { k } \). Find
(i) vector equation in parametric form
(ii) vector equation in non-parametric form
(iii) Cartesian equations of the straight line.

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a²=b²+c²−2bc cos A
(ii) b²=c²+a²−2ca cos B
(iii) c²= a²+b²−2ab cos C

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a=bcosC+ccos B
(ii) b=ccosA+acosC
(iii) c=acosB+bcos A