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)

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