Which is true from the following?
(A) Any vector \(\vec{r}\) in space can be written as \((\vec{r}\cdot\hat{\imath})\hat{\imath} + (\vec{r}\cdot\hat{\jmath})\hat{\jmath} + (\vec{r}\cdot\hat{k})\hat{k}\)
(B) If \(\vec{a}\) is perpendicular to \(\vec{b}\), \(|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2\)
(C) If \(|\vec{a}| = 2\), \(|\vec{b}| = 1\) and \(\vec{a}\cdot\vec{b} = 1\), the value of \((3\vec{a} - 5\vec{b})\cdot(2\vec{a} + 7\vec{b})\) is 1
(D) \(\vec{a} = 5\hat{\imath} - \hat{\jmath} - 3\hat{k}\) and \(\vec{b} = \hat{\imath} + 3\hat{\jmath} - 5\hat{k}\), the angle between \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\) is 60°
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) Only
For (A), let \(\vec{r} = x\hat{\imath} + y\hat{\jmath} + z\hat{k}\). Then
\(\vec{r}\cdot\hat{\imath} = x,\quad \vec{r}\cdot\hat{\jmath} = y,\quad \vec{r}\cdot\hat{k} = z\)
For (B), expanding with the dot product:
\(|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + 2\,\vec{a}\cdot\vec{b} + |\vec{b}|^2\)
For (C):
\((3\vec{a} - 5\vec{b})\cdot(2\vec{a} + 7\vec{b}) = 6|\vec{a}|^2 + 21\,\vec{a}\cdot\vec{b} - 10\,\vec{b}\cdot\vec{a} - 35|\vec{b}|^2\)
\(= 6(4) + 11(1) - 35(1) = 0\)
For (D):
\(\vec{a} + \vec{b} = 6\hat{\imath} + 2\hat{\jmath} - 8\hat{k}\)
\(\vec{a} - \vec{b} = 4\hat{\imath} - 4\hat{\jmath} + 2\hat{k}\)
\((\vec{a} + \vec{b})\cdot(\vec{a} - \vec{b}) = 24 - 8 - 16 = 0\)
So the components of \(\vec{r}\) are \(\vec{r}\cdot\hat{\imath}, \vec{r}\cdot\hat{\jmath}, \vec{r}\cdot\hat{k}\); for perpendicular vectors \(\vec{a}\cdot\vec{b} = 0\); the product in (C) is 0; and the angle in (D) is 90°.
Checking the statements
(A) Any vector \(\vec{r}\) in space can be written as \((\vec{r}\cdot\hat{\imath})\hat{\imath} + (\vec{r}\cdot\hat{\jmath})\hat{\jmath} + (\vec{r}\cdot\hat{k})\hat{k}\) — Correct. The three dot products are exactly x, y, z.
(B) If \(\vec{a}\) is perpendicular to \(\vec{b}\), \(|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2\) — Correct. The middle term \(2\,\vec{a}\cdot\vec{b}\) is 0.
(C) If \(|\vec{a}| = 2\), \(|\vec{b}| = 1\) and \(\vec{a}\cdot\vec{b} = 1\), the value of \((3\vec{a} - 5\vec{b})\cdot(2\vec{a} + 7\vec{b})\) is 1 — Incorrect. The value is 24 + 11 − 35 = 0.
(D) \(\vec{a} = 5\hat{\imath} - \hat{\jmath} - 3\hat{k}\) and \(\vec{b} = \hat{\imath} + 3\hat{\jmath} - 5\hat{k}\), the angle between \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\) is 60° — Incorrect. Their dot product is 0, so the angle is 90°.