PHAS0002: Mathematical Methods 1 — Problem Sheet 1
Week 7: Vectors
1. For the following pairs of vectors a, b, calculate
a + b,
a − b,
|a|,
|b|,
a · b,
a × b,
as well as the cosine of the angle between the two vectors:
(a) a = 3i + 5j − 7k, b = 2i + 7j + k,
(b) a = 2i + 3j, b = 3i + 3j,
(c) a = i + 2j + 3k, b = −3i − 6j − 9k,
(d) a = j + k, b = j − k,
2. Find the two unit vectors perpendicular to (i + j − k) and (2i − j + 3k) .
3. If a = i + 2j + k , b = −i + k and c = 3i + j − k , obtain
a · b , a × b , a · (b × c) and a × (b × c) .
4. Let a be an arbitrary vector and n̂ a unit vector pointing in an arbitrary direction. Show that a may be expressed as
a = (a · n̂) n̂ + (n̂ × a) × n̂ .
5. Find the angle between the position vectors to the points (3, −4, 0) and (−2, 1, 0)
and then find the direction cosines (i.e. the cosines of the angles with respect
to the i, j and k unit vectors) of a vector perpendicular to both of the above
position vectors.
6. The plane P1 contains the points A, B and C , which have position vectors
a = −3i + 2j, b = 7i + 2j and c = 2i + 3j + 2k respectively. Plane P2 passes
through A and is orthogonal to the line BC , whilst plane P3 passes through
B and is orthogonal to the line AC . Find the coordinates of r, the point of
intersection of the three planes.
The End