Match List I with List II
| LIST I | LIST II | ||
| A. | $|\vec{a}×\hat{i}|^2+|\vec{a}×\hat{j}|^2+|\vec{a}×\hat{k}|^2$=_________ | I. | $|\vec{a}|^2$ |
| B. | $\frac{(\vec{a}.\vec{b})^2+|\vec{a}×\vec{b}|^2}{|\vec{b}|^2}=$ | II. | $\sqrt{3}|\vec{a}|$ |
| C. | $\vec{a}, \vec{b}$ and $\vec{c}$ are three mutually perpendicular vectors of equal magnitude then $|\vec{a}+\vec{b}+\vec{c}|=$ | III. | $\frac{1}{2}|\vec{a}×\vec{b}|$ |
| D. | Area of paralleogram with diagonals $\vec{a}$ and $\vec{b}$ | IV. | $2|\vec{a}|^2$ |
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-IV, B-I,C-II, D-III
(A) $\vec a=a_x\hat i+a_y\hat j+a_z\hat k$
$|\vec{a}×\hat{i}|^2+|\vec{a}×\hat{j}|^2+|\vec{a}×\hat{k}|^2$
$=|-a_y\hat k+a_z\hat j|^2+|a_x\hat k-a_z\hat i|^2+|-a_x\hat j+a_y\hat i|^2$
$={a_y}^2+{a_z}^2+{a_x}^2+{a_z}^2+{a_x}^2+{a_y}^2$
$=2|\vec a|^2$ (IV)
(B) $\frac{(\vec{a}.\vec{b})^2+|\vec{a}×\vec{b}|^2}{|\vec{b}|^2}=\frac{|\vec{a}|^2|\vec{b}|^2(\cos^2θ+\sin^θ)}{|\vec{b}|^2}=|\vec{a}|^2$ (I)
(C) let $\vec a=a\hat i⇒b=b\hat j⇒c=c\hat k$
$a+b+c$
so $|\vec a+\vec b+\vec c|=|a\hat i+b\hat j+c\hat k|$
$=\sqrt{3}a=\sqrt{3}|\vec a|$ (II)
(D) area = $\frac{1}{2}|\vec{a}×\vec{b}|$ (III)