Download App

Scalar Or Dot Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSScalar Or Dot Product2 Marks
Question
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are three mutually perpendicular unit vectors, then prove that $\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|=\sqrt{3}$
✓

Answer

Given that $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ are unit vectors.
So, $\big|\vec{\text{a}}\big|=1,\Big|\vec{\text{b}}\big|=1$ and $\big|\vec{\text{c}}\big|=1$
Since they are mutually perpendicular,
$\vec{\text{a}}.\vec{\text{b}}=\vec{\text{b}}.\vec{\text{c}}=\vec{\text{c}}.\vec{\text{a}}=0$
Now,
$\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|^2=|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+|\vec{\text{c}}|+2\vec{\text{a}}.\vec{\text{b}}+2\vec{{\text{b}}}.\vec{\text{c}}+2\vec{\text{c}}.\vec{\text{a}}$
$=1+1+1+0+0+0$
$=3$
$\therefore\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|=\sqrt{3}$

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 "2 Marks Questions" from Scalar Or Dot Product→Scalar Or Dot Product — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Let $f : R - {-1} \rightarrow R - {1}$ be given by $\text{f(x)}=\frac{\text{x}}{\text{x}+1}.$ Write $f^{-1}(x).$
→
Events A and B are such that P(A) = $\frac{1}{2}$, P(B) = $\frac{7}{{12}}$ and P(not A or not B) = $\frac{1}{4}$. State whether A and B are independent?
→
Let $A$ and $B$ be two independent events such that $P(A) = p_1$ and $P(B) = p_2.$ Describe in words the events whose probabilities are:
$(1 - p_1)p_2.$
→
Verify that the function y - cos y = x (explicit or implicit) is a solution of differential equation (y sin y + cos y + x)y' = y
→
Evaluate the following integral:
$\int\frac{1}{\sqrt{\text{a}^2-\text{b}^2\text{x}^2}}\text{ dx}$
→
Find the position vector of a point R which divides the line segment joining points $\text{P}\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)$ and $\text{Q}\big(-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)$ in the ratio 2 : 1.Externally
→
Find values of k if area of triangle is 4 sq. units and vertices are: (-2, 0),(0, 4),(0, k).
→
A fair die is rolled. Consider events E = $\{1,\ 3,\ 5\},\ \text{F}=\{2,\ 3\}\ \text{and}\ \text{G}=\{2,\ 3,\ 4,\ 5\}.\ \text{Find}:$
$\text{P}(\text{E}|\text{G})\ \text{and}\ \text{P}(\text{G}|\text{E})$
→
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(1-\cos\theta)\text{ and y}=\text{a}(\theta+\sin\theta)\text{ at }\theta=\frac{\pi}{1}$
→
If $\begin{bmatrix}2\text{x}+1&5\text{x}\\0&\text{y}^2+1\end{bmatrix}=\begin{bmatrix}\text{x}+3&10\\0&26\end{bmatrix},$ find the value of (x + y).
→