Section 1.2: The Dot Product
Definition: The dot product of two vectors ~v = hv1 , v2 i and w
~ = hw1 , w2 i is the scalar
~v · w
~ = v1 w1 + v2 w2 .
Example: Find the dot product of the given vectors.
(a) ~v = h4, 7i, w
~ = h−2, 1i
(b) ~v = h−2, −8i, w
~ = h6, −4i
Theorem: (Angle Between Vectors)
If θ is the angle between two nonzero vectors ~v and w,
~ then
cos θ =
~v · w
~
.
||~v ||||w||
~
Note: The dot product of two vectors ~v and w
~ can also be defined as
~v · w
~ = ||~v ||||w||
~ cos θ,
where θ is the angle between ~v and w.
~
1
Example: Find the dot product ~v · w
~ given that ||~v || = 8, ||w||
~ = , and the angle between
2
π
the vectors ~v and w
~ is .
4
1
Example: Find the angle between the vectors ~v = h1, 2i and w
~ = h3, 4i.
Example: The points A = (3, 0), B = (5, 6), and C = (−2, 1) form a triangle. Find the angle
located at the vertex A.
Definition: Two vectors ~v and w
~ are parallel if there exists a scalar c such that ~v = cw.
~
Two vectors are orthogonal or perpendicular if the angle between them is θ = 90◦ .
Theorem: (Orthogonal Vector Theorem)
Two vectors ~v and w
~ are orthogonal if and only if
~v · w
~ = 0.
2
Example: Find all values of x such that ~v = hx, 2xi and w
~ = hx, −2i are orthogonal.
Example: Find all values of x such that ~v = h2, xi and w
~ = hx − 1, 3i are parallel.
Definition: The vector projection of ~v onto w
~ is the vector
~v · w
~
projw~ ~v =
w.
~
||w||
~ 2
The scalar projection or component of ~v onto w
~ is the scalar
~v · w
~
.
compw~ ~v =
||w||
~
Note: The scalar projection is the magnitude of the vector ~v acting in the direction of w.
~
3
Example: Find the vector and scalar projections of ~v = h3, −1i onto w
~ = h2, 3i.
Example: Find the distance from the point P = (2, 1) to the line y = 2x + 1.
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Applications to Physics and Engineering
Recall that the work done by a constant force F in moving an object a distance d is
W = F d. However, this formula only applies when the force acts in the direction of motion.
Suppose that a constant force F~ acts on an object moving from a point P to a point Q.
The work done in moving the object from P to Q depends on:
~
1. The distance the object has moved, ||D||.
2. The magnitude of the force applied in the direction of motion, ||F~ || cos θ.
The work done by F~ in moving the object is
~ cos θ = F~ · D.
~
W = ||F~ ||||D||
Example: A woman exerts a horizontal force of 30 lb on a crate as she pushes it up a ramp
that is 5 ft long and inclined at an angle of 20◦ above the horizontal. Find the work done
on the box.
5
Example: A boat sails south with the help of a wind blowing in the direction S36◦ E with
magnitude of 300 lb. Find the work done by the wind as the boat moves 150 ft.
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