Math 150 – Fall 2015
Section 9E
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Section 9E – Dot Product
Definition. The dot product (or scalar product) is an operation between two vectors
which gives a scalar answer. The dot product is defined as
hx1 , x2 i · hy1 , y2 i = x1 y1 + x2 y2 .
In Rn , the dot product is defined similarly,
hx1 , x2 , . . . , xn i · hy1 , y2 , . . . , yn i = x1 y1 + x2 y2 + · · · + xn yn .
Note. Don’t forget: the answer from a dot product is a NUMBER, not a VECTOR!
Example 1. Find h3, −7i · h5, −4i.
Example 2. Find h2, −4, −5i · h7, 1, 0i.
Theorem. For two vectors ~x and ~y , the lengths of the vectors, the angle θ between
them, and the dot product are related through the following formula:
~x · ~y = ||~x|| ||~y || cos θ
or
cos θ =
~x · ~y
||~x|| ||~y ||
where θ is the SMALLER of the two angles determined by the vectors ~x and ~y .
Example 3. Find the angle between the two vectors h4, 3i and h−2, 3i.
Example 4. Find the angle between the two vectors h2, −1, 5i and h−3, 2, −1i.
Math 150 – Fall 2015
Section 9E
2 of 2
Properties of the Dot Product
~ and Y
~ are
Theorem. The dot product of two vectors has the following properties. X
~
vectors of the same dimension, and X = hx1 , x2 , . . . , xn i.
~ ·X
~ = x2 + x2 + · · · + x2 = ||X||
~ 2.
1. X
n
1
2
~ ·Y
~ =Y
~ ·X
~
2. X
~ · Y
~ +Z
~ =X
~ ·Y
~ +X
~ · Z.
~
3. X
4.
~ ·Y
~ =X
~ · aY
~ =a X
~ ·Y
~
aX
Perpendicular Vectors
~ and Y
~ are perpendicular if and only if X
~ ·Y
~ = 0.
Theorem. Two vectors X
Example 5. Determine if the vectors h3, −5, 4i and h−12, 0, 9i are perpendicular.
Example 6. A man walks to 100 yards in a direction which is perpendicular to the
vector h2, 13i. Find a vector of length 100 which points in this direction.