Download App

VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
Find the angle between the vectors $\hat{i}-2\hat{j}+3\hat{k}\ \text{and}\ 3\hat{i}-2\hat{j}+\hat{k}.$
✓

Answer

$\text{Given:}\ \ \ \text{Let}\ \ \ \ \vec{a}=\hat{i}-2\hat{j}+3\hat{k}\ \text{and}\ \vec{b}=3\hat{i}-2\hat{j}+\hat{k}$ $\Rightarrow\ \ \big|\vec{a}\big|=\sqrt{1+4+9}=\sqrt{14}\ \text{and}\ \Big|\vec{b}\Big|=\sqrt{9+4+1}=\sqrt{14}$ $\Big[\because\ \text{x}\hat{i}+\text{y}\hat{j}+\text{z}\hat{k}=\sqrt{\text{x}^2+\text{y}^2+\text{z}^2}\ \Big]$ $\text{Also}\ \ \ \ \vec{a}.\vec{b}$= Product of coefficients of $\hat{i}$ + Product of coefficients of $\hat{j}$ + Product of coefficients $\hat{k}$
= 1(3) + (-2)(-2) + 3(1) = 3 + 4 + 3 = 10
Let $\theta$ be the angle between the vector $\vec{a}\ \text{and}\ \vec{b}.$ We know that $\text{cos}\ \theta=\frac{\vec{a}.\vec{b}}{\big|\vec{a}\big|.\big| \vec{b}\big|}$$\Rightarrow\ \ \text{cos}\ \theta=\frac{10}{\sqrt{14}\ .\sqrt{14}}=\frac{10}{14}=\frac{5}{7}$ $ \Rightarrow\ \text{cos}\ \theta=\frac{5}{7}$
$\Rightarrow\ \ \theta=\cos^{-1}\frac{5}{7}$

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 Question" 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

Prove that $\tan^{–1} \Bigg(\frac{\text{cos x}}{\text{1 + sinx}}\Bigg)=\frac{\pi}{4}-\frac{x}{2},\text{x}\in\Bigg(-\frac{\pi}{\text{2}},\frac{{\pi}}{2}\Bigg).$
→
Evaluate the following integrals:
$\int\frac{4\text{x}+3}{\sqrt{2\text{x}^2+3\text{x}+1}}\text{dx}$
→
Show that the normals to the following parirs of planes are perpendicular other.
x - y + z - 2 = 0 and 3x + 2y - z + 4 = 0
→
Given $\text{A}=\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix},$ compute $A^{-1}$ and show that $2A^{-1} = 9I - A$.
→
Check the commutativity and associativity of the following binary operations:
'*' on R defined by a * b = a + b - 7 for all a, b ∈ R.
→
Integrate the function $\frac{3 x}{1+2 x^{4}}$
→
$\int\frac{1}{2-3\text{x}}+\frac{1}{\sqrt{3\text{x}-2}}\text{dx}$
→
Calcualte the area under the curve $\text{y}=2\sqrt{\text{x}}$ included between the lines x = 0 and x = 1.
→
Find the value of $\lambda$ $\overrightarrow{a} ,\overrightarrow{b}$ and $\overrightarrow{c}$ coplanar, where $\overrightarrow{a} = 2\hat{i} -\hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} + 2\hat{j} - 3\hat{k} $ and $\overrightarrow{c} = 3\hat{i}- \lambda \hat{j} + 5\hat{k}.$
→
Find the values of k so that the function f is continuous at the indicated point:
$\text{f(x)}\begin{cases}\frac{\text{k}\cos\text{x}}{\pi -2\text{x}}\ \text{if}\ \text{x}\neq \frac{\pi}{2}\\3, \ \ \ \ \ \ \ \ \text{if}\ \text{x} =\frac{\pi}{2}\end{cases}$
$\text{at} \text{x} = \frac{\pi}{2}$
→