Pearson Edexcel IAL Mathematics FP3.5.1 Vector product & scalar triple product

Practise using vector products and scalar triple products to find perpendicular directions, parallelogram areas and volumes in 3D problems.

Exam points

calculate a×b to obtain a vector perpendicular to two given directions
use |a×b| or |a·(b×c)| to find an area or volume from given points

[Maximum number: 2]

The skew lines $l_{1}$ and $l_{2}$ have equations

l_{1}: \mathbf{r}=(\mathbf{i}+2 \mathbf{j}-5 \mathbf{k})+\lambda(5 \mathbf{i}+\mathbf{j})

and

l_{2}: \mathbf{r}=(2 \mathbf{i}-4 \mathbf{j}+4 \mathbf{k})+\mu(8 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k})

where $\lambda$ and $\mu$ are scalar parameters.

Determine a vector that is perpendicular to both $l_{1}$ and $l_{2}$

$(5\mathbf i+\mathbf j)\times(8\mathbf i-2\mathbf j+3\mathbf k)$
or solve
$(u\mathbf i+v\mathbf j+w\mathbf k)\cdot(5\mathbf i+\mathbf j)=0, \quad (u\mathbf i+v\mathbf j+w\mathbf k)\cdot(8\mathbf i-2\mathbf j+3\mathbf k)=0$ $5u+v=0,\quad 8u-2v+3w=0$ $\mathbf n=3\mathbf i-15\mathbf j-18\mathbf k$
or any non-zero multiple of $\mathbf i-5\mathbf j-6\mathbf k$ .

M1 A1

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FP3.5.1 - Vector product and scalar triple product question 1