Section 11.2: Vectors and the Dot Product in Three Dimensions
Definition: A vector in R3 is an ordered triple ~v = hv1 , v2 , v3 i of real numbers, where the
numbers v1 , v2 , v3 are called the components of ~v .
Note: As in R2 , vectors are represented as arrows with an initial and terminal point. The
vector with initial point A = (x1 , y1 , z1 ) and terminal point B = (x2 , y2 , z2 ) is given by
−→
AB = hx2 − x1 , y2 − y1 , z2 − z1 i.
A vector whose initial point is the origin is called a position vector.
Definition: The magnitude (length) of a vector ~v = hv1 , v2 , v3 i is given by
q
||~v || = v12 + v22 + v32 .
The vector with magnitude 0 is called the zero vector or null vector and is denoted by ~0.
Theorem: (Vector Algebra)
If ~v = hv1 , v2 , v3 i and w
~ = hw1 , w2 , w3 i are vectors and c is a scalar, then
(a) c~v = hcv1 , cv2 , cv3 i
(b) ~v + w
~ = hv1 + w1 , v2 + w2 , v3 + w3 i
(c) ~v − w
~ = hv1 − w1 , v2 − w2 , v3 − w3 i.
Vector Addition
𝑣+𝑤
𝑤
𝑤
𝑣
𝑣
Vector Subtraction
Scalar Multiplication
𝑣
𝑣−𝑤
𝑣
−𝑤
𝑐𝑣 , (0 < 𝑐 < 1)
𝑐𝑣 , (𝑐 > 1)
𝑐𝑣 , (𝑐 < 0)
Figure 1: Illustration of vector addition, subtraction, and scalar multiplication.
Definition: A unit vector is a vector with magnitude one. If ~v 6= ~0, then a unit vector in
the direction of ~v is given by
~v
~u =
.
||~v ||
Example: Let ~v = h−2, 4, 3i and w
~ = h1, 2, −1i. Find
(a) ~v + 3~b
~v + 3w
~ = h−2, 4, 3i + h3, 6, −3i = h1, 10, 0i.
(b) ||~v − w||
~
The difference is
~v − w
~ = h−3, 2, 4i.
Therefore,
||~v − w||
~ =
p
(−3)2 + 22 + 42 =
√
9 + 4 + 16 =
√
29.
(c) A unit vector in the direction of ~v .
The magnitude of ~v is
||~v || =
p
√
√
(−2)2 + 42 + 32 = 4 + 16 + 9 = 29.
So a unit vector in the direction of ~v is
~v
2
4
3
~u =
= −√ , √ , √
.
||~v ||
29 29 29
Definition: The unit vectors ~i = h1, 0, 0i, ~j = h0, 1, 0i, and ~k = h0, 0, 1i that point in the
directions of the positive x, y, and z axes are called the standard basis vectors.
Note: Any vector ~v = hv1 , v2 , v3 i can be expressed in terms of the standard basis vectors as
~v = v1~i + v2~j + v3~k.
Definition: The dot product of two vectors ~v = hv1 , v2 , v3 i and w
~ = hw1 , w2 , w3 i is
~v · w
~ = v1 w1 + v2 w2 + v3 w3 .
Example: Find the dot product of ~v = h2, 3, −1i and w
~ = h−3, 5, 4i.
~v · w
~ = 2(−3) + 3(5) + (−1)(4) = 5.
Theorem: (Angle Between Two Vectors)
If θ is the angle between two nonzero vectors ~v and w,
~ then
cos θ =
~v · w
~
.
||~v ||||w||
~
Example: Find the angle between the vectors ~v = h1, 2, 2i and ~v = h3, 4, 0i.
The dot product of the vectors is
~v · w
~ = 1(3) + 2(4) + 2(0) = 11.
The magnitudes of ~v and w
~ are
√
√
√
||~v || = 12 + 22 + 22 = 1 + 4 + 4 = 9 = 3,
√
√
√
||w||
~ = 32 + 42 + 02 = 9 + 16 + 0 = 25 = 5.
Therefore, cos θ =
11
and
15
−1
θ = cos
11
15
≈ 43◦ .
Definition: Two vectors are called orthogonal (perpendicular) if the angle between them is
θ = π/2 radians or 90◦ .
Theorem: (Orthogonal Vector Theorem)
Two nonzero vectors ~v and w
~ are orthogonal if and only if
~v · w
~ = 0.
Example: Determine whether the given vectors are orthogonal, parallel, or neither.
(a) ~v = h1, 5, −2i and w
~ = h3, 1, 4i
The vectors are orthogonal since
~v · w
~ = 1(3) + 5(1) − 2(4) = 0.
(b) ~v = h2, −1, 3i and w
~ = h−4, 2, −6i
The vectors are parallel since
w
~ = h−4, 2, −6i = −2h2, −1, 3i = −2~v .
(c) ~v = h1, 0, −2i and w
~ = h2, −1, 3i
The vectors are neither orthogonal nor parallel. They are not orthogonal since
~v · w
~ = 1(2) + 0 − 2(3) = −4 6= 0.
They are not parallel since there is no scalar c such that ~v = cw.
~
~ is the vector
Definition: The vector projection of ~v onto w
~v · w
~
projw~ ~v =
w.
~
||w||
~ 2
The scalar projection or component of ~v onto w
~ is the scalar
compw~ ~v =
𝑣
𝑣∙𝑤
𝑝𝑟𝑜𝑗𝑤 𝑣
𝑐 𝑤
= cos 𝜃 =
=
𝑣 𝑤
𝑣
𝑣
𝜃
𝑝𝑟𝑜𝑗𝑤 𝑣 = 𝑐𝑤
𝑝𝑟𝑜𝑗𝑤 𝑣 = 𝑐𝑤 =
~v · w
~
.
||w||
~
𝑐=
𝑤
𝑣∙𝑤
𝑤
𝑤 2
𝑣∙𝑤
𝑤 2
𝑐𝑜𝑚𝑝𝑤 𝑣 =
𝑣∙𝑤
𝑤
Figure 2: Illustration of the vector and scalar projections of ~v onto w.
~
Example: Find the vector and scalar projections of ~v = h2, 3, 5i onto w
~ = h2, −2, −1i.
The dot product is
~v · w
~ = 2(2) + 3(−2) + 5(−1) = 4 − 6 − 5 = −7.
The magnitude of w
~ is
||w||
~ =
p
√
22 + (−2)2 + (−1)2 = 9 = 3.
Thus, the vector and scalar projections of ~v onto w
~ are
7
14 14 7
projw~ ~v = − h2, −2, −1i = − , ,
,
9
9 9 9
7
compw~ ~v = − .
3