1. Dot Product and Angle Between Two Vectors

2. Given two vectors These vectors are perpendicular if the lines that contain the vectors have a slope whose product is -1.
Two vectors are parallel if the lines that contain them are parallel, or, we say that two vectors are parallel if
for some real number k.

3. The Dot Product or Scalar ProductIf
then the dot product of these vectors is defined by:
Example: If
find:

4. Properties of the Dot Product
Perpendicular vectors are sometimes called orthogonal vectors.

5. From the Cosine Rule:

7. The Angle Between Two Vectors

8. If , and
show that and are perpendicular and also show
that and are parallel.

9. If and find
Solution: 4
If and find the measure of the angle
between and
Solution: 0.644

10. If and find the measure of the
angle between and
Solution: 1.85
If the angle between the vectors and
is 45 degrees, find the two possible values of a.
Solution: -4 or 1

11. The points A, B, C and D have position vectors 5i + j, -3i + 2j,
-3i – 3j and i – 6j. Show that is perpendicular to
The points A, B, C and D have position vectors
and respectively where ,
, and If AC is perpendicular to BD, find the value of y.
Solution: 1

12. Given two non-zero vectors a and b such that
find the value of
Solution: 0
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