11th Maths Weekly Test -2 (Vector Algebra)

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test -2 (Vector Algebra)-Aug 2020

Time : 1.30 Hours

Part - A

Generic 2 - Answer All Question

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

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.

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{j}+4\hat{k}\)and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\). Find\((\vec{a}+\vec{b}).\vec{c}\) and \(\vec{c}.(\vec{a}+\vec{b})\) Are they equal?

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

Find the direction cosines and direction ratios for the following vectors \(\hat{j}\)

Find the vectors of magnitude 6 which are perpendicular to both the vectors \(4\vec { i } -\vec { j } +3\vec { k } \) and \(-2\vec { i } +\vec { j } -2\vec { k } \)

Find a direction ratio and direction cosines of the following vectors \(3\hat{i}+4\hat{j}-6\hat{k}\)

For any two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\), prove that \(|\overrightarrow{a}\times \overrightarrow{b}|^2+(\overrightarrow{a}. \overrightarrow{b})^2=|\overrightarrow{a}|^2 |\overrightarrow{b}|^2\)

If \(\vec{a}=3\vec{i}-2\vec{j}+\vec{k}\) and \(\vec{b}=2\vec{i}-4\vec{j}+\vec{k}\) then find \(\left| \vec { a } -2\vec { b } \right| \)

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 5 - Answer All Question

Find the projection of:
(i) \(\hat { i } -\hat { j } \) on Z-axis
(ii) \(\hat { i } +2\hat { j } -2\hat { k } \) on \(2\hat { i } -\hat { \quad j } -2\hat { k } \)
(iii) \(3\hat { i } +\hat { j } -\hat { k } \) on \(4\hat { i } -\hat { j } +2\hat { k } \)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)

Let ABC be a triangle,\(\overrightarrow{BC}=\overrightarrow{a},\overrightarrow{CA}=\overrightarrow{b}\) and \(\overrightarrow{AB}=\overrightarrow{c}\). Then prove that \(\overrightarrow {a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow {c}=\overrightarrow {c}\times \overrightarrow {a}.\)

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three vectors such that \(\left| \vec { a } \right| =3.\left| \vec { b } \right| =4\) and \(\left| \vec { c } \right| =\sqrt { 24 } \) sum of any two vectors is orthogonal to the third vector, then find \(\left| \vec { a } +\vec { b } +\vec { c } \right| \).

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11th Maths Weekly Test -2 (Vector Algebra)

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test -2 (Vector Algebra)-Aug 2020

Part - A

Generic 2 - Answer All Question

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

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.

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{j}+4\hat{k}\)and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\). Find\((\vec{a}+\vec{b}).\vec{c}\) and \(\vec{c}.(\vec{a}+\vec{b})\) Are they equal?

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

Find the direction cosines and direction ratios for the following vectors \(\hat{j}\)

Find the vectors of magnitude 6 which are perpendicular to both the vectors \(4\vec { i } -\vec { j } +3\vec { k } \) and \(-2\vec { i } +\vec { j } -2\vec { k } \)

Find a direction ratio and direction cosines of the following vectors \(3\hat{i}+4\hat{j}-6\hat{k}\)

For any two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\), prove that \(|\overrightarrow{a}\times \overrightarrow{b}|^2+(\overrightarrow{a}. \overrightarrow{b})^2=|\overrightarrow{a}|^2 |\overrightarrow{b}|^2\)

If \(\vec{a}=3\vec{i}-2\vec{j}+\vec{k}\) and \(\vec{b}=2\vec{i}-4\vec{j}+\vec{k}\) then find \(\left| \vec { a } -2\vec { b } \right| \)

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 5 - Answer All Question

Find the projection of: (i) \(\hat { i } -\hat { j } \) on Z-axis (ii) \(\hat { i } +2\hat { j } -2\hat { k } \) on \(2\hat { i } -\hat { \quad j } -2\hat { k } \) (iii) \(3\hat { i } +\hat { j } -\hat { k } \) on \(4\hat { i } -\hat { j } +2\hat { k } \)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a2 = b2 + c2 - 2bc cos A (Law of cosines)

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Find the projection of:
(i) \(\hat { i } -\hat { j } \) on Z-axis
(ii) \(\hat { i } +2\hat { j } -2\hat { k } \) on \(2\hat { i } -\hat { \quad j } -2\hat { k } \)
(iii) \(3\hat { i } +\hat { j } -\hat { k } \) on \(4\hat { i } -\hat { j } +2\hat { k } \)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)