A vector makes an angle of π 4 with each of x - axis and y - axis. Find the angle made by it with the z - axis.

Let the vector O P → makes an angle α = 45 ° and β = 45 ° with O X , O Y respectively. Suppose O P → is inclined at angle γ to O Z Let l , m , n be the direction cosines of O P → . Then, l = m = cos π 4 = 1 2 n = cos γ Now, we have, l 2 + m 2 + n 2 = 1 ⇒ 1 2 + 1 2 + n 2 = 1 ⇒ n 2 = 0 ⇒ n = 0 ⇒ cos γ = cos π 2 ⇒ γ = π 2 Hence, the angle made by it with the z - axis is π 2 .

A vector r → is inclined to x - axis at 45 ° and y - axis at 60 ° . If | r → | = 8 units , find r → .

Here, r → makes an angle 45 ° with O X and 60 ° with O Y . So, l = cos 45 ° = 1 2 and m = cos 60 ° = 1 2 Now, l 2 + m 2 + n 2 = 1 ⇒ 1 2 + 1 4 + n 2 = 1 ⇒ n 2 = 1 4 ⇒ n = ± 1 2 Therefore, r → = r → l i ^ + m j ^ + n k ^ = 8 1 2 i ^ + 1 2 j ^ ± 1 2 k ^ = 4 2 i ^ + j ^ ± k ^

If a unit vector a → makes an angle π 3 with i ^ , π 4 with j ^ and an acute angle θ with k ^ , then find θ and hence, the components of a → .

The Direction cosines of vector a → are l = cos π 3 = 1 2 , m = cos π 4 = 1 2 , n = cos θ Therefore, l 2 + m 2 + n 2 = 1 ⇒ 1 4 + 1 2 + cos 2 θ = 1 ⇒ cos θ = 1 2 ⇒ θ = π 3 [ ∵ a → makes acute angle with k ^ ] Since, a → is the unit vector. ∴ a → = l i ^ + m j ^ + n k ^ ⇒ a → = 1 2 i ^ + 1 2 j ^ + 1 2 k ^ Hence, components of a → are 1 2 i ^ + 1 2 j ^ + 1 2 k ^ .

Find a vector r → of magnitude 3 2 units which makes an angle of π 4 and π 4 with y and z - axes respectively.

Suppose vector r → makes an angle α with the x - axis. Let l , m , n be the direction cosines of r → . Then, l = cos α , m = cos π 4 = 1 2 , n = cos π 2 = 0 Now, l 2 + m 2 + n 2 = 1 ⇒ cos 2 α + 1 2 + 0 = 1 ⇒ cos α = ± 1 2 We know that r → = r → l i ^ + m j ^ + n k ^ ∴ r → = 3 2 ± 1 2 i ^ + 1 2 j + 0 k ^ r → = 3 2 ⇒ r → = ± 3 i ^ + 3 j ^ .

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE

R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 9: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 9: EXERCISE

Attempt the free practice questions on Chapter 4: Algebra of Vectors, Exercise 9: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-2 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 9: EXERCISE with Hints & Solutions

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 3

A vector makes an angle of $\Rightarrow γ = \frac{π}{2}$ with each of $x -$ axis and $y -$ axis. Find the angle made by it with the $z -$ axis.

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 4

A vector $\vec{r}$ is inclined at equal acute angles to $x -$ axis, $y -$ axis and $z -$ axis. If $|\vec{r}| = 6 units$ , find $\vec{r}$ .

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 5

A vector $\vec{r}$ is inclined to $x -$ axis at $45 °$ and $y -$ axis at $60 °$ . If $| \vec{r} | = 8 units$ , find $\vec{r}$ .

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 6

Find the direction cosines of the following vectors: $3 \hat{i} - 4 \hat{k}$

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 7

Find the angles at which the following vector is inclined to each of the coordinate axes:

$\hat{j} - \hat{k}$

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 10

If a unit vector $\vec{a}$ makes an angle $\frac{π}{3}$ with $\hat{i}, \frac{π}{4}$ with $\hat{j}$ and an acute angle $θ$ with $\hat{k}$ , then find $θ$ and hence, the components of $\vec{a}$ .