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IB Mathematics: Applications and Interpretation HL · AHL 4: Vectors · AHL 4.20

Mixed practice: scalar and vector products, lines and distances Problem set A Answers

Skills

1. (a) 120°
2. (b) 127°
1. (a) 53.1°
2. (b) 112°
3. (c) 3.26
4. (d) \(12 \sqrt{3}\)
5. (e) 0.848 rad
6. (f) 0.623 rad
1. (a) 63.6°
2. (b) 79.9°
3. (c) \(\mathbf{d}_1\cdot\mathbf{d}_2=0\)
4. (d) \(\mathbf{r}=\begin{pmatrix}2\\5\\1\end{pmatrix}+\lambda \begin{pmatrix}3\\0\\-4\end{pmatrix}\); 53.1°
1. (a) \(-5\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}\) (or any multiple of it)
2. (b) \(\dfrac{1}{\sqrt{6}}\begin{pmatrix}-1\\1\\2\end{pmatrix}\) (or its negative)
3. (c) \(t=-1\)
4. (d) t = -1 or t = 1.5
1. (a) \(\mathbf{a}\times\mathbf{b}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\), and the scalar product of perpendicular vectors is 0
2. (b) 0
3. (c) Both sides equal 126
4. (d) \(\mathbf{p}=\begin{pmatrix}-1\\8\\-5\end{pmatrix}\), \(\mathbf{q}=\begin{pmatrix}1\\-8\\5\end{pmatrix}\); \(\mathbf{q}=-\mathbf{p}\)

1. (a) 111°, 39.6°, 29.2°
2. (b) 10.4
1. (a) \(\overrightarrow{AB}\cdot\overrightarrow{AC}=0\)
2. (b) B: 54.0°, C: 36.0°
3. (c) 6.18
1. (a) (2, 5, 4)
2. (b) 17.3
1. (a) (5, 3, 3)
2. (b) \(|\overrightarrow{AB}|=|\overrightarrow{AD}|=3\)
3. (c) \(\overrightarrow{AC}\cdot\overrightarrow{BD}=0\)
1. (a) C(7, 4, 0), F(7, 1, 4), G(7, 4, 4), H(2, 4, 4)
2. (b) 13.9

1. (a) \(\overrightarrow{OB}=\lambda\overrightarrow{OA}\)
2. (b) \(\lambda=3\)
3. (c) \(\sqrt{2}\)
1. (a) M(5, 0, -1)
2. (b) Q is on \(l\) when \(\lambda=-3\)
3. (c) R(15, 5, -11)

jmaths.xyz · AI HL · AHL 4.20 Mixed practice: scalar and vector products, lines and distances