# Vectors – Further Pure Maths 1

Candidates should be able to:

• use the equation of a plane in any of the forms $$ax + by + cz = d$$ or $$\mathbf{r.n} = p$$ or $$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$$, and convert equations of planes from one form to another as necessary in solving problems;
• recall that the vector product $$\mathbf{a} × \mathbf{b}$$ of two vectors can be expressed either as $$\mathbf{|a|} \mathbf{|b|} \sin \theta \mathbf{\hat{n}}$$ , where $$\mathbf{\hat{n}}$$ is a unit vector, or in component form as $$(a_2b_3 – a_3b_2) \mathbf{i} + (a_3b_1 – a_1b_3) \mathbf{j} + (a_1b_2 – a_2b_1) \mathbf{k}$$;
• use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections, including:
• determining whether a line lies in a plane, is parallel to a plane or intersects a plane, and finding the point of intersection of a line and a plane when it exists,
• finding the perpendicular distance from a point to a plane,
• finding the angle between a line and a plane, and the angle between two planes,
• finding an equation for the line of intersection of two planes,
• calculating the shortest distance between two skew lines,
• finding an equation for the common perpendicular to two skew lines.

Vectors – Further Pure Maths 1 Video (4:11:54)