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 \(\vec{PQ} = -3\hat{\imath} - 5\hat{\jmath} - 4\hat{k}\)
(B) Projection of a vector \(\vec{a}\) on other vector \(\vec{b}\) is \(\displaystyle\frac{\vec{a}.\vec{b}}{|\vec{a}|}\)
(C) If \(\vec{a} = \hat{\imath} - 2\hat{\jmath} + \hat{k}\) and \(\vec{b} = -2\hat{\imath} + 4\hat{\jmath} + 5\hat{k}\) then \(\vec{a} + \vec{b} = -\hat{\imath} + 2\hat{\jmath} + 6\hat{k}\)
(D) If \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\) then \(\cos\theta = \displaystyle\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}\)
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
The vector from P to Q is the position vector of Q minus that of P:
\( \vec{PQ} = (-1-2)\hat{\imath} + (-2-3)\hat{\jmath} + (-4-0)\hat{k} \)
\( = -3\hat{\imath} - 5\hat{\jmath} - 4\hat{k} \)
The projection of \(\vec{a}\) on \(\vec{b}\) divides by the length of \(\vec{b}\), the vector on which it is projected:
\( \displaystyle \text{projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \)
Adding the components:
\( \vec{a} + \vec{b} = (1-2)\hat{\imath} + (-2+4)\hat{\jmath} + (1+5)\hat{k} \)
\( = -\hat{\imath} + 2\hat{\jmath} + 6\hat{k} \)
From the definition of the dot product, \(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta\):
\( \displaystyle \cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|} \)
So \(\vec{PQ} = -3\hat{\imath} - 5\hat{\jmath} - 4\hat{k}\), the projection on \(\vec{b}\) has \(|\vec{b}|\) in the denominator, \(\vec{a} + \vec{b} = -\hat{\imath} + 2\hat{\jmath} + 6\hat{k}\), and \(\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\).
Checking the statements
(A) The vector joining the points \(P(2, 3, 0)\) and \(Q(-1,-2,-4)\) directed from P to Q is \(\vec{PQ} = -3\hat{\imath} - 5\hat{\jmath} - 4\hat{k}\) — Correct. Q minus P gives −3, −5, −4.
(B) Projection of a vector \(\vec{a}\) on other vector \(\vec{b}\) is \(\frac{\vec{a}.\vec{b}}{|\vec{a}|}\) — Incorrect. The denominator should be \(|\vec{b}|\), not \(|\vec{a}|\).
(C) If \(\vec{a} = \hat{\imath} - 2\hat{\jmath} + \hat{k}\) and \(\vec{b} = -2\hat{\imath} + 4\hat{\jmath} + 5\hat{k}\) then \(\vec{a} + \vec{b} = -\hat{\imath} + 2\hat{\jmath} + 6\hat{k}\) — Correct. The components add to −1, 2, 6.
(D) If \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\) then \(\cos\theta = \frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}\) — Correct. This is the dot-product formula for the angle.