Seminar 3 Mathematics for Engineerings Dr.
Nan Meng
1. Find the vector equation of the line with cartesian equation
x+2 y+1 z−1
= = .
3 2 −1
2. If L1 is the line through the point with cartesian coordinates (1, 1, 3) parallel to the vector
(1, −2, 2) and L2 is the line through the point (4, 0, −1) parallel to the vector (3, −2, −2),
find the coordinates of the point where L1 and L2 intersect.
3. Find the angle between the planes
x + 3y + 2z = 7,
−2x − y + 3z = −1,
and find the perpendicular distance of the point A(1, 1, 1) from each plane.
4. Find the vector and cartesian equations of the plane
(a) which is perpendicular to p = (1, 3, −4) and passes through the point with position
vector (2, −1, −1),
(b) which contains the points (1, 0, 1), (2, 1, 1) and (3, 2, 4),
(c) which is parallel to the vectors (1, 0, −1) and (2, −1, 1) and which passes through
the point (−1, 2, −1).
5. Find the equation of the plane which passes through the points (2, 2, 5), (3, 4, 6), (2, 0, 2).
6. Find the vector and cartesian equations of the straight line through the points with
position vectors (−3, 2, −1) and (4, 0, −3).
Find the vector and cartesian equation of the plane containing this line and the point
(−1, 1, 0).
7. (a) Find the cartesian equation of the straight line in two dimensions which passes
through the points A and B with coordinates (0, 1) and (1, 4). If the points A and
B have position vectors a and b, show that the vector equation of the line can be
expressed in the parametric form
r = a + s(b − a), −∞ < s < ∞,
where r = xi + yj and s is a measure of distance along the line from A. Write down
the vector equation of the line AB defined above in terms of x, y, s and the basis
vectors i and j by inserting the numerical values for the vectors a and b. By equating
components and then eliminating s, show that this reproduces the cartesian formula
derived earlier.
(b) The points C and D have three-dimensional coordinates (1, 2, 2) and (2, 3, 4), re-
spectively. Calculate the vector CD and hence write down the vector equation of
the line passing through these points. Show that the point (3, 4, 6) lies on this line.
Show also that the cartesian equation of the line CD is
x−1 y−2 z−2
= = .
1 1 2
