Download App

Vector or Cross Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product5 Marks
Question
If $\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}},$ then verify that $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)=\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{a}}\times\vec{\text{c}}.$
✓

Answer

We have,
$\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}}$
$\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$
$\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}},$
$\big(\vec{\text{b}}+\vec{\text{c}}\big)=(\text{b}_1+\text{c}_1)\hat{\text{i}}+(\text{b}_2+\text{c}_2)\hat{\text{j}}+(\text{b}_3+\text{c}_3)\hat{\text{k}}$
Now, $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{a}_2&\text{a}_3\\\text{b}_1+\text{c}_{1}&\text{b}_2+\text{c}_2&\text{b}_3+\text{c}_3 \end{vmatrix}$
$=\hat{\text{i}}\big[\text{a}_2(\text{b}_3+\text{c}_3)-\text{a}_3(\text{b}_2+\text{c}_2)\big]-\hat{\text{j}}\big[\text{a}_1(\text{b}_3+\text{c}_3)-\text{a}_3(\text{b}_1+\text{c}_1)\big]\\+\hat{\text{k}\big[\text{a}}_1(\text{b}_2+\text{c}_2)-\text{a}_2(\text{b}_1+\text{c}_1)\big]$
$=\hat{\text{i}}\big[\text{a}_2\text{b}_3+\text{a}_2\text{c}_3-\text{a}_3\text{b}_2-\text{a}_3\text{c}_2\big]+\hat{\text{j}}\big[-\text{a}_1\text{b}_3-\text{a}_1\text{c}_3+\text{a}_3\text{b}_1+\text{a}_3\text{c}_1\big]\\+\hat{\text{k}}\big[\text{a}_1\text{b}_2+\text{a}_1\text{c}_2-\text{a}_2\text{b}_1-\text{a}_2\text{c}_1\big]\dots(1)$
$$$\vec{\text{a}}\times\vec{\text{b}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{a}_2&\text{a}_3\\\text{b}_1&\text{b}_2&\text{b}_3 \end{vmatrix}$
$=\hat{\text{i}}\big[\text{a}_2\text{b}_3-\text{a}_3\text{b}_2\big]+\hat{\text{j}}\big[\text{b}_1\text{a}_3-\text{a}_1\text{b}_3\big]+\hat{\text{k}}\big[\text{a}_1\text{b}_2-\text{a}_2\text{b}_1\big]\ \dots(2)$
$\vec{\text{a}}\times\vec{\text{c}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{a}_2&\text{a}_3\\\text{c}_1&\text{c}_2&\text{c}_3 \end{vmatrix}$
$=\hat{\text{i}}\big[\text{a}_2\text{c}_3-\text{a}_3\text{c}_2\big]+\hat{\text{j}}\big[\text{a}_3\text{c}_1-\text{a}_1\text{c}_3\big]+\hat{\text{k}}\big[\text{a}_1\text{c}_2-\text{a}_3\text{c}_1\big]\ \dots(3)$

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 "Solve the Following Question.(5 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 intervals in which $\text{f}(\text{x})=\log(1+\text{x})-\frac{\text{x}}{1+\text{x}}$ is increasing or decreasing.
→
If $\text{xy}=1,$ prove that $\frac{\text{dy}}{\text{dx}}+\text{y}^2=0$
→
Find the area bounded by the parabola $y^2 = 4x$ and the line $y = 2x - 4:$
By using horizontal strips.
→
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at $8\%$ per year, compounded continuously. Calculate the percentage increase in such an account over one year.
→
Using vector method, prove that the point is collinear:
A(6, -7, -1), B(2, -3, 1) and C(4, -5, 0)
→
Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$
→
Find the particular solution of the differential equation $\text{x}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\frac{\text{dy}}{\text{dx}}=\text{y}\cos\Big(\frac{\text{y}}{\text{x}}\Big)+\text{x},$ given that when $\text{x}=1,\text{y}=\frac{\pi}4$.
→
Find the vector equation of the line passing through the points (-1, 0, 2) and (3, 4, 6).
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = {x(x - 2)}^2$​​​​​​​
→
Show that the vectors
$\vec{\text{a}}=\frac{1}{7}(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}),\vec{\text{b}}=\frac{1}{7}(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}),\vec{\text{c}}=\frac{1}{7}(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})$ are mutually perpendicular unit vectors.
→