Express a Vector in Component Form
Find the component form of
with initial point
A(1, –3) and terminal point B(1, 3).
= 〈x2 – x1, y2 – y1〉
Component form
= 〈1 – 1, 3 – (–3)〉
(x1, y1 ) = (1, –3)
and ( x2, y2 ) = (1, 3)
= 〈0, 6〉
Subtract.
Answer:
〈0, 6〉
Find the component form of
given initial point
A(–4, –3) and terminal point B(5, 3).
Find the Magnitude of a Vector
Find the magnitude of
with initial point A(1, –3)
and terminal point B(1, 3).
Magnitude formula
(x1 , y1 ) = (1, –3) and
( x2 , y2 ) = (1, 3)
Simplify.
Find the Magnitude of a Vector
Answer: 6
CHECK From Example 1, you know that
9
= 〈0, 6〉.
Find the magnitude of
given initial point
A(4, –2) and terminal point B(–3, –2).
Operations with Vectors
A. Find 2w + y for w = 〈2, –5〉, y = 〈2, 0〉, and
z = 〈–1, –4〉.
2w + y = 2 〈2, –5〉 + 〈2, 0〉
= 〈4, −10〉 + 〈2, 0〉
Substitute.
Scalar
multiplication
= 〈4 + 2, –10 + 0〉 or 〈6, –10〉 Vector addition
Answer:
〈6, –10〉
Operations with Vectors
B. Find 3y – 2z for w = 〈2, –5〉, y = 〈2, 0〉, and
z = 〈–1, –4〉.
3y – 2z = 3y + (–2z)
Answer:
Rewrite subtraction
as addition.
= 3〈2, 0〉 + (–2)〈–1, –4〉
Substitute.
= 〈6, 0〉 + 〈2, 8〉
Scalar multiplication
= 〈6 + 2, 0 + 8〉 or 〈8, 8〉
Vector addition
〈8, 8〉
Find 3v + 2w for v = 〈4, –1〉 and w = 〈–3, 5〉.
Vectors are denoted with
bold letters
v = < a, b >
This is the notation for a
position vector. This means
the point (a, b) is the
terminal point and the initial
point is the origin.
v = ai + bj
(a, b)
j
i
We use vectors that are only 1
unit long to build position
vectors. i is a vector 1 unit long
in the x direction and j is a vector
1 unit long in the y direction.
(3, 2)
j
j
v = < 3, 2 >
i i i
v = 3i + 2 j
If we want to add vectors that are in the form ai + bj, we can
just add the i components and then the j components.
v = −2i + 5 j
w = 3i − 4 j
v + w = − 2i + 5 j + 3i − 4 j = i + j
Let's look at this geometrically:
Can you
see from
this picture
5j
how to find
the length
of v?
3i
w
v
− 2i i
− 4j
j
When we want to know
the magnitude of the
vector (remember this is
the length) we denote it
v
=
(− 2) + (5)
2
= 29
2
A unit vector is a vector with magnitude 1.
If we want to find the unit vector having the same
direction as a given vector, we find the magnitude of the
vector and divide the vector by that value.
w = 3i − 4 j
w =
( 3) + ( −4 )
2
What is w ?
2
= 25 = 5
If we want to find the unit vector having the same direction
as w we need to divide w by 5.
3 4
u= i− j
5 5
Let's check this to see if it really is
1 unit long.
2
2
25
⎛3⎞ ⎛ 4⎞
=1
u = ⎜ ⎟ +⎜− ⎟ =
25
⎝5⎠ ⎝ 5⎠
Use this formula to find the a unit vector
Unit vector with the same
direction as v.
Find a Unit Vector with the Same Direction as
a Given Vector
Find a unit vector u with the same direction as
v = 〈4, –2〉.
Unit vector with the same
direction as v.
Substitute.
; Simplify.
or
Find a Unit Vector with the Same Direction as
a Given Vector
Rationalize the denominator.
Scalar multiplication
Rationalize denominators.
Therefore, u =
Answer: u =
.
Find a Unit Vector with the Same Direction as
a Given Vector
Check Since u is a scalar multiple of v, it has the same
direction as v. Verify that the magnitude of u is 1.
Magnitude
Formula
Simplify.
9
Simplify.
Find a unit vector u with the same direction as
w = 〈5, –3〉.
Write a Unit Vector as a Linear Combination of
Unit Vectors
Let
be the vector with initial point D(–3, –3) and
terminal point E(2, 6). Write
as a linear
combination of the vectors i and j.
First, find the component form of
.
= 〈x2 – x1, y2 – y1〉
Component form
= 〈2 – (–3), 6 – (–3)〉
(x1 , y1 ) = (–3, –3)
and ( x2 , y2 ) = (2, 6)
= 〈5, 9〉
Subtract.
Write a Unit Vector as a Linear Combination of
Unit Vectors
Then, rewrite the vector as a linear combination of the
standard unit vectors.
= 〈5, 9〉
Component form
= 5i + 9j
〈a, b〉 = ai + bj
Answer: 5i + 9j
Let
be the vector with initial point D(–4, 3) and
terminal point E(–1, 5). Write
as a linear
combination of the vectors i and j.
Find Component Form
Find the component form of the vector v with
magnitude 7 and direction angle 60°.
Component form of v in terms
of |v| and θ
|v| = 7 and θ = 60°
Simplify.
Answer:
Find Component Form
Check Graph v =
≈ 〈3.5, 6.1〉.
The measure of the angle v makes with the
positive x-axis is about
60° as shown, and
|v| =
.9
Find the component form of the vector v with
magnitude 12 and direction angle 300°.
Direction Angles of Vectors
A. Find the direction angle of p = 〈2, 9〉 to the
nearest tenth of a degree.
Direction angle equation
a = 2 and b = 9
Solve for θ.
Use a calculator.
Direction Angles of Vectors
So the direction angle of vector p is about 77.5°, as
shown below.
Answer: 77.5°
Direction Angles of Vectors
B. Find the direction angle of r = –7i + 2j to the
nearest tenth of a degree.
Direction angle equation
a = –7 and b = 2
Solve for θ.
Use a calculator.
Direction Angles of Vectors
Since r lies in Quadrant II as shown below,
θ = 180 – 15.9° or 164.1°.
Answer: 164.1°
Find the direction angle of p = 〈–1, 4〉 to the nearest
tenth of a degree.
Applied Vector Operations
SOCCER A soccer player running forward at 7
meters per second kicks a soccer ball with a
velocity of 30 meters per second at an angle of 10°
with the horizontal. What is the resultant speed
and direction of the kick?
Since the soccer player moves straight forward, the
component form of his velocity v1 is 〈7, 0〉. Use the
magnitude and direction of the soccer ball’s velocity v2
to write this vector in component form.
Applied Vector Operations
v 2 = 〈| v2 | cos θ, | v2 | sin θ〉
Component form of v2
= 〈30 cos 10°, 30 sin 10°〉
|v2| = 30 and θ = 10°
≈ 〈29.5, 5.2〉
Simplify.
Add the algebraic vectors representing v1 and v2 to
find the resultant velocity, r.
r = v1 + v2
Resultant vector
= 〈7, 0〉 + 〈29.5, 5.2〉
Substitution
= 〈36.5, 5.2〉
Vector Addition
Applied Vector Operations
The magnitude of the resultant is |r| =
or about 36.9. Next find the resultant direction θ.
Applied Vector Operations
〈a, b〉 = 〈36.5, 5.2〉
Therefore, the resultant velocity of the kick is about
36.9 meters per second at an angle of about 8.1° with
the horizontal.
Answer: 36.9 m/s; 8.1°
SOCCER A soccer player running forward at 6
meters per second kicks a soccer ball with a
velocity of 25 meters per second at an angle of 15°
with the horizontal. What is the resultant speed
and direction of the kick?