5.Ans: $|\vec{a}| = \sqrt{x^2 + y^2 + z^2}$
6.Ans: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$
7.Ans: $\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$
8.Ans: $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$
10.Ans: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$
11.Ans: $\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}$
13.Ans: Collinear (or Parallel)
14.Ans: $x = 3, y = 2, z = -1$
15.Ans: $\vec{AB} = (4-1)\hat{i} + (5-2)\hat{j} + (6-3)\hat{k} = 3\hat{i} + 3\hat{j} + 3\hat{k}$
16.Ans: Direction Cosines ($l, m, n$)
19.Ans: Magnitude is $\sqrt{1^2+2^2+2^2} = 3$. Direction cosines are $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$.
20.Ans: $\vec{r} = \frac{m\vec{q} + n\vec{p}}{m+n}$
21.Ans: $\vec{r} = \frac{m\vec{q} - n\vec{p}}{m-n}$
22.Ans: $\frac{\vec{p} + \vec{q}}{2}$
23.Ans: $3\hat{i} + 4\hat{j}$
24.Ans: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta$
29.Ans: $\vec{a} \cdot \vec{b} = 0$
30.Ans: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$
31.Ans: $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$
32.Ans: $(2)(1) + (1)(-3) = 2 - 3 = -1$
33.Ans: $(3)(1) + (-2)(1) + (1)(-1) = 3 - 2 - 1 = 0$
35.Ans: $0$ (Since $\hat{i}$ and $\hat{j}$ are perpendicular)
36.Ans: $\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \, \hat{n}$
38.Ans: Right-Hand Thumb Rule
39.Ans: $\vec{0}$ (Zero vector)
43.Ans: False ($\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$)
44.Ans: $\vec{a} \times \vec{b} = \vec{0}$
46.Ans: Area = $|\vec{a} \times \vec{b}|$
47.Ans: Area = $\frac{1}{2} |\vec{a} \times \vec{b}|$
48.Ans: Area = $\frac{1}{2} |\vec{d_1} \times \vec{d_2}|$
49.Ans: $[\vec{a} \, \vec{b} \, \vec{c}]$
50.Ans: $\vec{b} \times \vec{c}$
52.Ans: $[\vec{a} \, \vec{b} \, \vec{c}] = 0$
54.Ans: $1$ (Since $\hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1$)
55.Ans: $0$ (Two vectors are identical)
56.Ans: $(\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$
57.Ans: No, it is generally not associative.
58.Ans: $(\vec{a} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{a})\vec{b}$
