OCR A-Level Further Maths A further vectors and planes overview quiz quiz

a vector perpendicular to both of the inputs

a number (scalar)

always zero for any two vectors

a matrix of three rows and one column

$\mathbf{a}\times\mathbf{b} = 0$

$|\mathbf{a}| = |\mathbf{b}|$

$\mathbf{a}\cdot\mathbf{b} = 0$

$\mathbf{a} = \mathbf{b}$

a number equal to $|\mathbf{a}||\mathbf{b}|\cos\theta$ between the two vectors

a vector parallel to $\mathbf{a}$ with magnitude $|\mathbf{a}||\mathbf{b}|$

the angle $\theta$ measured between the two vectors $\mathbf{a}$ and $\mathbf{b}$

a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$

the parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$

the triangle whose two sides are the vectors $\mathbf{a}$ and $\mathbf{b}$ themselves

a unit square of side one, regardless of the two vectors $\mathbf{a}$ and $\mathbf{b}$

a circle of radius $|\mathbf{a}|$ drawn about the origin of the two vectors

$\mathbf{r}\cdot\mathbf{n} = d$

$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$

$ax + by + cz = d$

$\mathbf{r} = \lambda\mathbf{a}\times\mathbf{b}$

coincident

perpendicular

skew

parallel

stop at once, because two non-parallel lines in 3D must always intersect somewhere

compute the cross product of the two direction vectors to locate the point of crossing

find the midpoint of the two position vectors, which gives the point where they meet

test the third component for consistency

$\mathbf{r}\cdot\mathbf{n} = d$

$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$

$\mathbf{r}\times\mathbf{n} = d$

$|\mathbf{r}| = d$

$2\mathbf{i}$

$3\mathbf{i} - \mathbf{j} + 4\mathbf{k}$

$\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$

$3\mathbf{i} + \mathbf{j} + 4\mathbf{k}$

$\cos\theta = \frac{|\mathbf{d}\cdot\mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$

$\tan\theta = \mathbf{d}\cdot\mathbf{n}$

$\sin\theta = \frac{|\mathbf{d}\cdot\mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$

$\theta = \mathbf{d}\times\mathbf{n}$

$\frac{|ax_0 + by_0 + cz_0 - d|}{a^2 + b^2 + c^2}$

$\frac{ax_0 + by_0 + cz_0 - d}{\sqrt{a^2 + b^2 + c^2}}$

$\frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}$

$\frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$

the dot product of their normals

the cross product of a point taken from each of the two planes in turn

the sum of their $d$-values divided by the magnitudes of the two normals

the difference between the areas enclosed by the two planes being compared