Which of the following statements are true?
(A) The vector joining the points $P(2, 3, 0)$ and $Q(-1,-2,-4)$ directed from P to Q is $\vec{PQ} = -3\hat i - 5\hat j - 4\hat k$
(B) Projection of a vector $\vec a$ on other vector $\vec b$ is $\frac{\vec a.\vec b}{|\vec a|}$
(C) If $\vec a =\hat i-2\hat j + \hat k$ and $\vec b = -2\hat i + 4\hat j+ 5\hat k$ then $\vec a+\vec b=-\hat i+2\hat j+6\hat k$
(D) If is the angle between $\vec a$ and $\vec b$ then $\cos θ =\frac{\vec a.\vec b}{|\vec a||\vec b|}$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
(A) The vector from P(2, 3, 0) to Q(−1, −2, −4) is:
$\vec{PQ} = (-1 - 2)\hat{i} + (-2 - 3)\hat{j} + (-4 - 0)\hat{k} = -3\hat{i} -5\hat{j} -4\hat{k}$ → True
(B) Projection of $\vec{a}$ on $\vec{b}$ is given by:
$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ → So this statement is False
(C) $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$
Then, $\vec{a} + \vec{b} = (-1)\hat{i} + 2\hat{j} + 6\hat{k}$ → True
(D) Cosine of angle between $\vec{a}$ and $\vec{b}$ is given by:
$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$ → True