11th Maths Monthly Test - 3 ( Vector Algebra )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 3 ( Vector Algebra )-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Find the area of the parallelogram whose adjacent sides are \(\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}\) and \(\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\) .

Find \(\overrightarrow{a}.\overrightarrow{b}\) when \(\overrightarrow{a}\) and \(\overrightarrow{b}\) represent the points (2, 3, - 1) and (- 1, 2, 3).

Find \(|\overrightarrow{a}\times \overrightarrow{b}|,\)where \(\overrightarrow{a}=3\hat{i}+4\hat{j}\)and\(\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}\).

Find the direction cosines of a vector whose direction ratios are 0,0,7

Verify whether the following ratios are direction cosines of some vector or not\({4\over 3}.0,{3\over 4}\)

If \({1\over2},{1\over \sqrt{2}}\),a are the direction cosines of some vector, then find a.

Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\)-3\(\hat{k}\)+4\(\hat{j}\)

If \(\left| \vec { a } \right| =2,\) ,\(\left| \vec { b } \right| =7\) and \(\vec { a } \times \vec { b } =3\hat { i } -2\hat { j } +6\hat { k } \) find the angle between \(\vec { a } \) and \(\vec { b } \)

Verify whether the following ratios are direction cosines of some vector or not \({1\over 5},{3\over 5},{4\over 5}\)

Show that the vectors \(3\hat { i } -2\hat { j } +\hat { k } \) ,\(\hat { i } -3\hat { j } +5\hat { k } \) and \(2\hat { i } +\hat { j } -4\hat { k } \) form a right angled triangle.

Classify the following as scalar and vector quantities. (i) time period (ii) distance (iii) force (iv) velocity (v) work done.

Find the direction cosines of the line joining (2, 3, 1) and (3, - 1, 2).

Find the angle between the vectors \(\hat{i}-\hat{j}\)and\(\hat{j}-\hat{k}\).

Find the vectors of magnitude \(10\sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2\hat{j}+\hat{k}\) and\(\hat{i}+3\hat{j}+4\hat{k}\)

Part - B

Generic 3 - Answer All Question

Show that the vectors 2\(\hat{i}\)−\(\hat{j}\)+3\(\hat{k}\),3 \(\hat{i}\) −4\(\hat{j}\) -4\(\hat{k}\), \(\hat{i}\) −3\(\hat{j}\) -5 \(\hat{k}\) form a right angled triangle.

Show that the points (4, - 3, 1), (2, - 4, 5) and (1, - 1, 0) form a right angled triangle.

If \(\overrightarrow{a}=2\hat{i}+3\hat{j}-4\hat{k},\) \(\overrightarrow{b}=3\hat{i}-4\hat{j}-5\hat{k},\) and \(\overrightarrow{c}=-3\hat{i}+2\hat{j}+3\hat{k},\)find the magnitude and direction cosines of \(\overrightarrow{a}\)+ \(\overrightarrow{b}\) + \(\overrightarrow{c}\)

Show that the vectors \(-\hat{i}-2\hat{j}-6\hat{k},2\hat{i}-\hat{j}+\hat{k}\) ,and \(-\hat{i}+3\hat{j}+5\hat{k}\) form a right angled triangle.

Find the value of \(\lambda\) for which the vectors \(\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k} \) and \(\overrightarrow{b}=\overrightarrow{i}+\lambda \overrightarrow{j}+3\overrightarrow{k}\) are parallel.

Find the area of the parallelogram whose two adjacent sides are determined by the vectors\(\hat{i}+2\hat{j}+3\hat{k}\) and \(3\hat{i}-2\hat{j}+\hat{k}\)

Find the unit vectors parallel to the sum of \(3\vec { i } -5\vec { j } +8\vec { k } \) and \(-2\vec { i } -2\vec { k } \)

Let \(\vec { u } ,\vec { v } \) and \(\vec { w } \) be vectors such that \(\vec { u } +\vec { v } +\vec { w } =\vec { o } \) .If \(\vec { \left| u \right| } =3\) ,\(\left| \vec { v } \right| =4\) and \(\left| \vec { w } \right| =5\) then find \(\vec { u } .\vec { v } +\vec { v } .\vec { w } .\vec { u } \)

If \(\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\)\(\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}\) and \(\overrightarrow{c}=3\hat{i}+\hat{j}\) be such that \(\overrightarrow{a}+\lambda \overrightarrow{b}\) is perpendicular to \(\overrightarrow{c}\) then find \(\lambda\).

Find the area of the triangle whose vertices are A(3, - 1, 2), B(1, - 1, - 3) and C(4, - 3, 1).

Show that the points whose positions vectors \(4\hat { i } -3\hat { j } +\hat { k } \) ,\(2\hat { i } -4\hat { j } +5\hat { k } \) ,\(\hat { i } -\hat { j } \) from a right angled triangle.

Show that the points whose position vectors 4\(\hat{i}\) +5\(\hat{j}\) + \(\hat{k}\) , -\(\hat{j}\) -\(\hat{k}\) , 3\(\hat{i}\)+9\(\hat{j}\) +4 \(\hat{k}\) and -4\(\hat{i}\) +4\(\hat{j}\) +4\(\hat{k}\) are coplanar.

For any vector \(\overrightarrow{r}\) prove that \(\overrightarrow{r}\) = (\(\overrightarrow{r}.\hat{i}\)) \(\hat{i}\)+ (\(\overrightarrow{r}.\hat{j}\)) \(\hat{j}\)+ + (\(\overrightarrow{r}.\hat{k}\)) \(\hat{k}\).

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

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11th Maths Monthly Test - 3 ( Vector Algebra )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 3 ( Vector Algebra )-Aug 2020

Part - A

Generic 2 - Answer All Question

Find the area of the parallelogram whose adjacent sides are \(\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}\) and \(\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\) .

Find \(\overrightarrow{a}.\overrightarrow{b}\) when \(\overrightarrow{a}\) and \(\overrightarrow{b}\) represent the points (2, 3, - 1) and (- 1, 2, 3).

Find \(|\overrightarrow{a}\times \overrightarrow{b}|,\)where \(\overrightarrow{a}=3\hat{i}+4\hat{j}\)and\(\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}\).

Find the direction cosines of a vector whose direction ratios are 0,0,7

Verify whether the following ratios are direction cosines of some vector or not\({4\over 3}.0,{3\over 4}\)

If \({1\over2},{1\over \sqrt{2}}\),a are the direction cosines of some vector, then find a.

Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\)-3\(\hat{k}\)+4\(\hat{j}\)

If \(\left| \vec { a } \right| =2,\) ,\(\left| \vec { b } \right| =7\) and \(\vec { a } \times \vec { b } =3\hat { i } -2\hat { j } +6\hat { k } \) find the angle between \(\vec { a } \) and \(\vec { b } \)

Verify whether the following ratios are direction cosines of some vector or not \({1\over 5},{3\over 5},{4\over 5}\)

Show that the vectors \(3\hat { i } -2\hat { j } +\hat { k } \) ,\(\hat { i } -3\hat { j } +5\hat { k } \) and \(2\hat { i } +\hat { j } -4\hat { k } \) form a right angled triangle.

Classify the following as scalar and vector quantities. (i) time period (ii) distance (iii) force (iv) velocity (v) work done.

Find the direction cosines of the line joining (2, 3, 1) and (3, - 1, 2).

Find the angle between the vectors \(\hat{i}-\hat{j}\)and\(\hat{j}-\hat{k}\).

Find the vectors of magnitude \(10\sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2\hat{j}+\hat{k}\) and\(\hat{i}+3\hat{j}+4\hat{k}\)

Part - B

Generic 3 - Answer All Question

Show that the vectors 2\(\hat{i}\)−\(\hat{j}\)+3\(\hat{k}\),3 \(\hat{i}\) −4\(\hat{j}\) -4\(\hat{k}\), \(\hat{i}\) −3\(\hat{j}\) -5 \(\hat{k}\) form a right angled triangle.

Show that the points (4, - 3, 1), (2, - 4, 5) and (1, - 1, 0) form a right angled triangle.

If \(\overrightarrow{a}=2\hat{i}+3\hat{j}-4\hat{k},\) \(\overrightarrow{b}=3\hat{i}-4\hat{j}-5\hat{k},\) and \(\overrightarrow{c}=-3\hat{i}+2\hat{j}+3\hat{k},\)find the magnitude and direction cosines of \(\overrightarrow{a}\)+ \(\overrightarrow{b}\) + \(\overrightarrow{c}\)

Show that the vectors \(-\hat{i}-2\hat{j}-6\hat{k},2\hat{i}-\hat{j}+\hat{k}\) ,and \(-\hat{i}+3\hat{j}+5\hat{k}\) form a right angled triangle.

Find the value of \(\lambda\) for which the vectors \(\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k} \) and \(\overrightarrow{b}=\overrightarrow{i}+\lambda \overrightarrow{j}+3\overrightarrow{k}\) are parallel.

Find the area of the parallelogram whose two adjacent sides are determined by the vectors\(\hat{i}+2\hat{j}+3\hat{k}\) and \(3\hat{i}-2\hat{j}+\hat{k}\)

Find the unit vectors parallel to the sum of \(3\vec { i } -5\vec { j } +8\vec { k } \) and \(-2\vec { i } -2\vec { k } \)

If \(\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\)\(\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}\) and \(\overrightarrow{c}=3\hat{i}+\hat{j}\) be such that \(\overrightarrow{a}+\lambda \overrightarrow{b}\) is perpendicular to \(\overrightarrow{c}\) then find \(\lambda\).

Find the area of the triangle whose vertices are A(3, - 1, 2), B(1, - 1, - 3) and C(4, - 3, 1).

Show that the points whose positions vectors \(4\hat { i } -3\hat { j } +\hat { k } \) ,\(2\hat { i } -4\hat { j } +5\hat { k } \) ,\(\hat { i } -\hat { j } \) from a right angled triangle.

Show that the points whose position vectors 4\(\hat{i}\) +5\(\hat{j}\) + \(\hat{k}\) , -\(\hat{j}\) -\(\hat{k}\) , 3\(\hat{i}\)+9\(\hat{j}\) +4 \(\hat{k}\) ​and ​​​​​​ -4\(\hat{i}\) +4\(\hat{j}\) +4\(\hat{k}\) ​are coplanar.​​​​​​

For any vector \(\overrightarrow{r}\) prove that \(\overrightarrow{r}\) = (\(\overrightarrow{r}.\hat{i}\)) \(\hat{i}\)+ (\(\overrightarrow{r}.\hat{j}\)) \(\hat{j}\)+ + (\(\overrightarrow{r}.\hat{k}\)) \(\hat{k}\).

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

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Show that the points whose position vectors 4\(\hat{i}\) +5\(\hat{j}\) + \(\hat{k}\) , -\(\hat{j}\) -\(\hat{k}\) , 3\(\hat{i}\)+9\(\hat{j}\) +4 \(\hat{k}\) and -4\(\hat{i}\) +4\(\hat{j}\) +4\(\hat{k}\) are coplanar.