Which of the following statements are correct?
A. The unit vector in the direction of the vector \( \vec{a}=\hat{\imath}+\hat{\jmath}+2\hat{k} \) is \( \displaystyle\frac{1}{\sqrt{6}}\hat{\imath}+\displaystyle\frac{1}{\sqrt{6}}\hat{\jmath}+\displaystyle\frac{2}{\sqrt{6}}\hat{k} \).
B. A vector in the direction of the vector \( 5\hat{\imath}-\hat{\jmath}+2\hat{k} \) which has magnitude 8 units is \( 40\hat{\imath}-8\hat{\jmath}+16\hat{k} \).
C. The vector joining the points P(2, 3, 0) and Q(−1, −2, −4) directed from P to Q is \( -3\hat{\imath}-5\hat{\jmath}-4\hat{k} \).
D. The position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, −2) is \( 3\hat{\imath}+2\hat{\jmath}-\hat{k} \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A and C only
For the vector \( \hat{\imath}+\hat{\jmath}+2\hat{k} \):
\( \displaystyle |\vec{a}|=\sqrt{1+1+4}=\sqrt{6} \)
\( \displaystyle \hat{a}=\frac{\vec{a}}{|\vec{a}|} \)
\( \displaystyle =\frac{1}{\sqrt{6}}\hat{\imath}+\frac{1}{\sqrt{6}}\hat{\jmath}+\frac{2}{\sqrt{6}}\hat{k} \)
A vector of magnitude 8 in a given direction is 8 times the unit vector, not 8 times the vector:
\( \displaystyle |5\hat{\imath}-\hat{\jmath}+2\hat{k}|=\sqrt{25+1+4}=\sqrt{30} \)
\( \displaystyle \text{Required vector}=\frac{8}{\sqrt{30}}\left(5\hat{\imath}-\hat{\jmath}+2\hat{k}\right) \)
From P to Q:
\( \displaystyle \overrightarrow{PQ}=(-1-2)\hat{\imath}+(-2-3)\hat{\jmath}+(-4-0)\hat{k} \)
\( \displaystyle =-3\hat{\imath}-5\hat{\jmath}-4\hat{k} \)
Mid-point of P(2, 3, 4) and Q(4, 1, −2):
\( \displaystyle \frac{\overrightarrow{OP}+\overrightarrow{OQ}}{2}=\frac{6\hat{\imath}+4\hat{\jmath}+2\hat{k}}{2} \)
\( \displaystyle =3\hat{\imath}+2\hat{\jmath}+\hat{k} \)
So the unit vector is \( \frac{1}{\sqrt{6}}(\hat{\imath}+\hat{\jmath}+2\hat{k}) \), the vector of magnitude 8 is \( \frac{8}{\sqrt{30}}(5\hat{\imath}-\hat{\jmath}+2\hat{k}) \), \( \overrightarrow{PQ}=-3\hat{\imath}-5\hat{\jmath}-4\hat{k} \) and the mid-point is \( 3\hat{\imath}+2\hat{\jmath}+\hat{k} \).
Checking the statements
(A) The unit vector in the direction of the vector \( \vec{a}=\hat{\imath}+\hat{\jmath}+2\hat{k} \) is \( \frac{1}{\sqrt{6}}\hat{\imath}+\frac{1}{\sqrt{6}}\hat{\jmath}+\frac{2}{\sqrt{6}}\hat{k} \). — Correct. \( |\vec{a}|=\sqrt{6} \).
(B) A vector in the direction of the vector \( 5\hat{\imath}-\hat{\jmath}+2\hat{k} \) which has magnitude 8 units is \( 40\hat{\imath}-8\hat{\jmath}+16\hat{k} \). — Incorrect. The given vector has magnitude \( 8\sqrt{30} \), not 8.
(C) The vector joining the points P(2, 3, 0) and Q(−1, −2, −4) directed from P to Q is \( -3\hat{\imath}-5\hat{\jmath}-4\hat{k} \). — Correct. Q − P = (−3, −5, −4).
(D) The position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, −2) is \( 3\hat{\imath}+2\hat{\jmath}-\hat{k} \). — Incorrect. The \( \hat{k} \) component is (4 − 2)/2 = +1, not −1.