# 6. 1. the diagram below.

```Section 7.4 Extra Practice
STUDENT BOOK PAGES 379–387
Copyright &copy; 2009 by Nelson Education Ltd.
1. For each of the following pairs of vectors, determine
both their dot product and the angle, to the nearest
degree, between the
&gt; two vectors.
&gt;
a. a ⫽ (⫺1, 2), b ⫽ (1, 2)
&gt;
&gt;
b. a ⫽ (3, 4), b ⫽ (8, ⫺6)
&gt;
&gt;
c. a ⫽ (1, 1, ⫺1), b ⫽ (2, 1, ⫺6)
&gt;
&gt;
d. a ⫽ (2, ⫺2, 1), b ⫽ (1, ⫺1, 2)
&gt;
2. For what
&gt; values of k are the vectors a ⫽ (1, 1, ⫺1)
and b ⫽ (2k, 2, ⫺3k) perpendicular?
&gt;
&gt;
3. For the vectors p ⫽ (k, ⫺2, ⫺1) and q ⫽ (k, 1, k):
a. are there any values of k for which the vectors are
perpendicular?
b. are there any values of k for which the vectors
are collinear?
&gt;
&gt;
4. The vectors p ⫽ (1, 2) and q ⫽ (⫺1, 0) form two
sides of a parallelogram.
a. Sketch the parallelogram.
b. Determine the vector components for the two
diagonals of the parallelogram.
c. Determine the interior angles of the parallelogram.
5. The triangle ABC has vertices A(⫺1, 2), B(3, 2), and
C(1, 4).
a. Determine the angles within the triangle.
b. Show that the sum of the angles of this triangle
is 180&deg;.
6. A rectangular box measuring 4 by 4 by k is shown in
the diagram below.
z
F
E
D
P(4, 4, k)
C
O
x
A
y
B
a. If k ⫽ 4, what is the length of the line segment AE?
b. What is the value of k for which the line segment
AE has length 10?
7. Find the components of a vector perpendicular to the
&gt;
&gt;
vectors p ⫽ (1, 2, ⫺1) and q ⫽ (⫺1, 0, 2).
8. Consider the two points A(⫺1, 1, 0) and B(1, 2, 3).
a. Determine the components of the vector from
A to B.
b. Find a vector perpendicular to the vector in part a.
c. Use the vector from part b. to find the coordinates
of a point C such that 䉭ABC is a right triangle.
9. Use the dot product formula to show that if
&gt; &gt;
&gt;
p ⴢ p ⫽ 0 for the vector p ⫽ (a, b, c), then
a ⫽ b ⫽ c ⫽ 0.
Section 7.4 Extra Practice
433
```