Examples on Vector Algebra

Comp.145  October 1996  Dr. A.J. Mestel

1. (a) Find the scalar product of the two vectors (2, x, 1)^{T} and (4, −2, −2)^{T}. What value does x have if they are perpendicular?

(b) Determine the angle between the vectors (2, 2, −1)^{T} and (6, −3, 2)^{T}.

2. A three-dimensional object is rotated about an axis through the origin with an angular velocity vector (1, 1, 1)^{T}. Show that a vertex whose coordinates are (1, 1, 0) moves with a velocity (−1, 1, 0)^{T}.

3. When \underline{a} = (2, −3, −1)^{T} and \underline{b} = (1, 4, −2)^{T} find

(i) \underline{a} ∧ \underline{b}  (ii) \underline{b} ∧ \underline{a}  (iii) (\underline{a} + \underline{b}) ∧ (\underline{a} − \underline{b})

Simplify (iii) by expanding the brackets, and check that your answers for (i), (ii) and (iii) are consistent.

4. Find, in vector form, the equation of the line through the points A and B, whose coordinates are (1, −2, 1) and (2, −2, 3) respectively. Hence determine the position of the shadow of an object at A due to a light at B on a screen in the plane x = 0.

5. Find the equation of the plane through the points (1, 1, 0), (2, −2, −1) and (0, 5, 2), by first finding two vectors which lie in the plane and then finding one which must be perpendicular to it.

6. A tetrahedron is a pyramid-like three-dimensional figure, formed by joining each of four points to all the others. It is important in various Finite Element Methods of computation.

The vertices of a tetrahedron, A,B,C and D, have position vectors \underline{a}, \underline{b}, \underline{c} and \underline{d} respectively. Find the position vector of the midpoint of the line joining the midpoints of the lines AB and CD. Do the same for the midpoints of AC and BD, and then for AD and BC.

Deduce that the lines joining the midpoints of opposite edges of any tetrahedron all meet at a single point, and that they bisect each other.