Find the angle between the vectors 2 i - j + k and 3 i +4 j - k . - Vidyadip

Question

Find the angle between the vectors $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $3\hat{\text{i}}+4\hat{\text{j}}-\hat{\text{k}}.$

✓

Answer

Let $\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=3\hat{\text{i}}+4\hat{\text{j}}-\hat{\text{k}}$
We know that, angle between two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ is given by
$\cos\theta=\frac{\vec{\text{a}}\cdot\vec{\text{b}}}{|\vec{\text{a}}||\vec{\text{b}}|}$
$=\frac{(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})(3\hat{\text{i}}+4\hat{\text{j}}-\hat{\text{k}})}{\sqrt{4+1+1}\sqrt{9+16+1}}$
$=\frac{6-4-1}{\sqrt{6}\sqrt{26}}=\frac{1}{2\sqrt{39}}$
$\therefore\theta=\cos^{-1}\Big(\frac{1}{2\sqrt{39}}\Big)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from VECTOR ALGEBRA→VECTOR ALGEBRA — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Discuss the continuity of $\text{f}\text{(x)}=\begin{cases}2\text{x}-1, & \text{x} < 0\\2\text{x}+1, & \text{x} \geq 0\end{cases}\text{at}\text{ x}=0$

Find the direction cosines of the vector joining the points A(1, 2, -3) and B(-1, -2, 1) directed from A to B.

If $\text{y}=\log_\text{a}\text{x},$, find $\frac{\text{dy}}{\text{dx}}.$

Find the value of $\int \frac{\sin x}{\sin (x-a)} d x$.

By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\frac{\cos^{5}\text{x}\ \text{dx}}{\sin^{5}\text{x}+\cos^{5}\text{x}}$

If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda$ so that $\text{A}^2 = 5\text{A} + \lambda\text{I}.$

Prove the following results:
$2\tan^{-1}\frac{3}{4}-\tan^{-1}\frac{17}{31}=\frac{\pi}{4}$

Evalute the following integrals:
$\int\frac{1+\tan\text{x}}{1-\tan\text{x}}\text{dx}$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

$\text{y}=\text{e}^{\text{x}}(\text{a}\cos\text{x}+\text{b}\sin\text{x})$

$\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$

Evaluate the following integrals:
$\int\frac{1}{1-\sin\text{x}}\text{dx}$