Problems on Vectors
problem 1
Three vectors a, b, and c, each having a magnitude of 50 units, lie in the xy plane and
make angles of 30°, 195°, and 315° with the positive x axis, respectively. Find
graphically the magnitudes and directions of the vectors (a) a + b + c, (b) a - b + c, and
(c) a vector d such that (a + b) - (c + d) = 0.
problem 2
A vector B, when added to the vector C = 3.0i + 4.0j, yields a resultant vector that is in
the positive y direction and has a magnitude equal to that of C. What is the magnitude of
B?
problem 3
Two vectors a and b have equal magnitudes of 10.0 units. They are oriented as shown in
the figure below and their vector sum is r. Find (a) the x and y components of r, (b) the
magnitude of r, and (c) the angle r makes with the positive x axis.
problem 4
Prove that two vectors must have equal magnitudes if their sum is perpendicular to their
difference.
problem 5
Use the definition of scalar product, a·b = ab cos θ, and the fact that a·b = axbx + ayby +
azbz to calculate the angle between the two vectors given by a = 3.0i + 3.0j + 3.0k and b
= 2.0i + 1.0j + 3.0k.
problem 6
Two vectors a and b have the components, in arbitrary units, ax = 3.2, ay = 1.6, bx = 0.50,
by = 4.5. (a) Find the angle between the directions of a and b. (b) Find the components of
a vector c that is perpendicular to a, is in the xy plane, and has a magnitude of 5.0 units.
problem 7
Vector a lies in the yz plane 63° from the +y axis, has a positive z component, and has
magnitude 3.20 units. Vector b lies in the xz plane 48° from the +x axis, has a positive z
component, and has magnitude 1.40 units. Find (a) a·b, (b) a×b, and (c) the angle
between a and b.
problem 8
Show that the area of the triangle contained between the vectors a and b and the red line
in the figure below is ½ │ a×b│.
problem 9
Show that a· (b × c) is equal in magnitude to the volume of the parallelepiped formed on
the three vectors a, b, and c as shown in the figure below.
problem 10
The three vectors shown in the figure below have magnitudes a = 3.00, b = 4.00, and c =
10.0. (a) Calculate the x and y components of these vectors. (b) Find the numbers p and q
such that c = pa + qb.