S-Cool Revision Summary

S-Cool Revision Summary

  1. Use Pythagoras to find the magnitude (length) of a vector.

  2. The direction of a vector is the angle measured anti-clockwise from the +ve x-axis.

  3. A position vector is the vector from the origin to a point.

  4. Given the magnitude (r) and direction (q) of a vector, the coordinates of a point are (r cosq, r sinq).

  5. Parallel vectors are multiples of each other.

  6. Perpendicular vectors have a scalar product of 0.

  7. Angles between vectors are found using:

Straight lines

A vector equation for a line is r = a + lb.

r = a general point on the line, a = a known point on the line, b = the direction of the line.

The Cartesian form is found by writing the x, y and z coordinates in terms of l.

Two lines intersect when their coordinates match. Find values of l and m that make all the coordinates match. If two lines do not meet then they are skew.

The angle between two lines = angle between their direction vectors.

Planes

The vector equation of a plane is:

, where a is a vector to the plane, and b and c are vectors in the plane.

The Cartesian equation of a plane is found using:

r.n = a.n, where a is a point in the plane and n is the normal to the plane.

To find where a line meets a plane:

  1. Write down the coordinates of a general point on the line.

  2. Use these coordinates in the equation of the plane.

  3. Solve the resulting equation and then find the coordinates.

To find the distance that a point is from a plane:

  1. Use the normal of the plane to make the equation of the line joining the point to the corresponding point on the plane.

  2. Find where this line meets the plane.

  3. Find the vector joining these two points.

  4. Use Pythagoras to find the length of this vector.

The shortest distance from the origin to a plane is found by dividing the constant in the cartesian equation by the length of the normal vector. (i.e. using the unit normal vector).

The angle between two planes is the same as the angle between the normals to the planes.

The angle between a line and a plane is 90o - angle between line and normal. Alternatively use:

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where a = direction of the line, and n = normal vector.