Download App

Vector or Cross Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product2 Marks
Question
For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}},$ find $\vec{\text{a}}.\big(\vec{\text{b}}\times\vec{\text{a}}\big).$
✓

Answer

Let:
$\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}}$
$\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$
$\vec{\text{b}}\times\vec{\text{a}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{b}_1&\text{b}_2&\text{b}_3\\\text{a}_1&\text{a}_2&\text{a}_3\end{vmatrix}$
$=\hat{\text{i}}(\text{b}_2\text{a}_3-\text{b}_3\text{a}_2)-\hat{\text{j}}(\text{b}_1\text{a}_3-\text{b}_3\text{a}_1)+\hat{\text{k}}(\text{b}_1\text{a}_2-\text{b}_2\text{a}_1)$
Now,
$\vec{\text{a}}.\big(\vec{\text{b}}\times\vec{\text{a}}\big)$
$=\big(\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}}\big).\big[\hat{\text{i}}(\text{b}_2\text{a}_3-\text{b}_3\text{a}_2)\\-\hat{\text{j}}(\text{b}_1\text{a}_3-\text{b}_3\text{a}_1)+\hat{\text{k}}(\text{b}_1\text{a}_2-\text{b}_2\text{a}_1)\big]$
$=\text{a}_1(\text{b}_2\text{a}_3-\text{b}_3\text{a}_2)-\text{a}_2(\text{b}_1\text{a}_3-\text{b}_3\text{a}_1)+\text{a}_3(\text{b}_1\text{a}_2-\text{b}_2\text{a}_1)$
$=\text{a}_1\text{b}_2\text{a}_3-\text{a}_1\text{b}_3\text{a}_2-\text{a}_2\text{b}_1\text{a}_3+\text{a}_2\text{b}_3\text{a}_1+\text{a}_3\text{b}_1\text{a}_2-\text{a}_3\text{b}_2\text{a}_1$
$=0$

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 "Solve the Following Question.(2 Marks)" from Vector or Cross Product→Vector or Cross Product — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the vector equation of the line passing through $2 \hat{i}+\hat{j}-\hat{k}$ and parallel to the line joining points $-\hat{i}+\hat{j}+4 \hat{k}$ and $\hat{i}+2 \hat{j}+2 \hat{k}$.
→
Find the mean of the following probability distribution:
$\text{X}=\text{x}_\text{i}:$ $1$ $2$ $3$
$\text{P}(\text{X}=\text{x}_\text{i}):$ $\frac{1}{4}$ $\frac{1}{8}$ $\frac{5}{8}$
→
Prove that $\tan ^{-1} 1+\tan ^{-1} 2+\tan ^{-1} 3=\pi$
→
Let * be a binary operation, on the set of all non-zero real numbers, given by
$\text{a}\times\text{b}=\frac{\text{ab}}{5}\ \forall\text{ a, b}\in\text{R}-\{0\}$
Write the value of x given by 2 * (x * 5) = 10.
→
Form the differential equation from the following primitives where constants are arbitrart:$\text{y}=\text{ax}^2+\text{bx}+\text{c}$
→
Evaluate the following:
$\text{cosec}^{-1}\Big(\text{cosec}\frac{11\pi}{6}\Big)$
→
If $\text{A}=\begin{bmatrix}1&2\\3&-1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\-1&0\end{bmatrix},$ find |AB|.
→
Evaluate the following : $\int \sin ^4 x \cdot d x$
→
Write the equation of the plane passing through (2, −1, 1) and parallel to the plane 3x + 2y − z = 7.
→
Find $a_{\text {, if }}$ the sum of the slopes of the lines represented by $a x^2+8 x y+5 y^2=0$ is twice their product.
→