In three‑dimensional space let $\mathbf{a} = (2,\,-1,\,3)$ and $\mathbf{b} = (k,\,4,\,-2)$. Let $\mathbf{c} = \mathbf{a}\times\mathbf{b}$. It is given that $|\mathbf{c}| = 13$ and that the vector $\mathbf{d} = \mathbf{a} + 2\mathbf{b}$ is orthogonal to $\mathbf{c}$. If $k

Question

In three‑dimensional space let $\mathbf{a} = (2,\,-1,\,3)$ and $\mathbf{b} = (k,\,4,\,-2)$. Let $\mathbf{c} = \mathbf{a}\times\mathbf{b}$. It is given that $|\mathbf{c}| = 13$ and that the vector $\mathbf{d} = \mathbf{a} + 2\mathbf{b}$ is orthogonal to $\mathbf{c}$. If $k < -1$, find the exact value of $k$ and then give the angle between $\mathbf{a}$ and $\mathbf{b}$ to the nearest degree.

Accepted Answer

k = -2 - \frac{\sqrt{290}}{10}; angle ≈ 143°

Igcse Model Questions Vol1 2023 Mathematics Question 15

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