Further vectors (Further Pure 1)

Explanation

• The vector product of $\mathbf{a}$ and $\mathbf{b}$ is the vector $\mathbf{a}\times\mathbf{b}=|\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{\mathbf{n}}$, where $\theta$ is the angle between them and $\hat{\mathbf{n}}$ is the unit vector perpendicular to both in the right-handed sense.
• In components it is the formal determinant with $\mathbf{i},\mathbf{j},\mathbf{k}$ in the first row and the components of $\mathbf{a}$ and $\mathbf{b}$ in the second and third.
• Two consequences do most of the work in examinations: $|\mathbf{a}\times\mathbf{b}|$ is the area of the parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$, so half of it is the area of the corresponding triangle; and $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ with both vectors non-zero means they are parallel.
• The product is anticommutative, $\mathbf{a}\times\mathbf{b}=-\mathbf{b}\times\mathbf{a}$, and it is not associative, so bracket every triple product carefully.

Explanation

• The scalar triple product $\mathbf{a}\cdot\left(\mathbf{b}\times\mathbf{c}\right)$ is the $3\times3$ determinant whose rows are the components of $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.
• Its modulus is the volume of the parallelepiped with those three edges, and one sixth of its modulus is the volume of the tetrahedron on the same three edges.
• Because swapping two rows of a determinant changes only its sign, the product is unchanged by a cyclic rotation, $\mathbf{a}\cdot\left(\mathbf{b}\times\mathbf{c}\right)=\mathbf{b}\cdot\left(\mathbf{c}\times\mathbf{a}\right)=\mathbf{c}\cdot\left(\mathbf{a}\times\mathbf{b}\right)$, and the dot and the cross may be interchanged.
• The product is zero exactly when the three vectors are coplanar, which is the standard test for coplanarity and the standard route to finding an unknown component that makes three vectors lie in one plane.
• Always take the modulus before quoting a volume; a negative determinant only records the handedness of the three edges.

Explanation

• A line through $\mathbf{a}$ with direction $\mathbf{b}$ can be written as $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$, in Cartesian form $\dfrac{x-a_1}{b_1}=\dfrac{y-a_2}{b_2}=\dfrac{z-a_3}{b_3}$, or in the vector-product form $\left(\mathbf{r}-\mathbf{a}\right)\times\mathbf{b}=\mathbf{0}$, which simply says that $\mathbf{r}-\mathbf{a}$ is parallel to $\mathbf{b}$.
• The components $b_1:b_2:b_3$ are the direction ratios, and dividing by $|\mathbf{b}|$ gives the direction cosines, whose squares sum to $1$.
• A plane is $\mathbf{r}\cdot\mathbf{n}=d$, with $\mathbf{n}$ obtained from a vector product when three points or a line and a point are given.
• The standard results follow from these: the angle between a line and a plane satisfies $\sin\theta=\dfrac{|\mathbf{b}\cdot\mathbf{n}|}{|\mathbf{b}||\mathbf{n}|}$; the distance from a point to a plane is $\dfrac{|\mathbf{p}\cdot\mathbf{n}-d|}{|\mathbf{n}|}$; and the shortest distance between skew lines uses the scalar triple product with the common perpendicular $\mathbf{u}\times\mathbf{v}$.