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Trigonometry Vector Math
Multiplying a Vector by a Scalar
A scalar is a number not a vector. In 3X, the number 3 is a scalar and means multiply X by three. In vectors, a scalar also means multiply.
If v^ G = 3 i − 2 j , then 3 v G = 3( 3 i − 2 j ), which is 9 i − 6 j.
A scalar lets you shrink or enlarge a vector like a photocopier. The vector has the same starting point and amplitude, but it has a new magnitude and ending point.
Example: If a^ G = 2 i − j , then 2 a G = 4 i − 2 j.
Example: If , then
G
b = − 10 i + 4 j^1 2
G
b = − 5 i + 2 j.
Example: What is the magnitude of 3 c G if c G = − 3 i + 4 j?
The vector 3 c G = − 9 i + 12 j. The magnitude is (−9) 2 + (12) 2 = 15.
If the scalar is a negative number, then the resulting vector has the opposite amplitude.
Example: If a^ G = 2 i − j , then− a G = − 2 i + j.
Adding Vectors Algebraically
When adding vectors, only like terms may be combined. This means we combine terms with terms and
G
G^ i i
G
j terms with
G
j terms. The sum of the vectors is called the resultant vector.
( − 10 i + 4 j )+ ( 3 i + j ) = − 7 i + 5 j
When subtracting vectors, we add the opposite.
( − 10 i + 4 j )− ( 3 i + j )
(− 10 i^ +^4 j ) + −( 3 i^ + −^ j ) = −^13 i^ +^3 j
Adding opposite vectors gives a zero vector (magnitude zero).
( 3 i + 2 j )+ (− 3 i + − 2 j )= 0 i + 0 j
Use the same process when adding or subtracting column vectors.
If m^ G = and
⎠⎟^
n G = 2 1
⎠⎟^
, then m G + n G =
and m G − n G =
5 + _^2
4 + _^1
⎠⎟^
This example involves scalar multiplication and vector addition/subtraction.
If and
G
b = − 10 i + 4 j c G = − 3 i + 4 j , then find 2
G
b − c
G
G
b − c G = 2 (− 10 i + 4 j ) + ( 3 i + − 4 j )
= − 20 i + 8 j + 3 i − 4 j = − 17 i + 4 j
Vectors on a Coordinate Plane Vectors are line segments in space, they have direction and length. That means we can show them on a coordinate grid. The vector v G = 2 i + 3 j can be drawn at many starting points. The vector is only the “road map” or directions to go from the starting point to the ending point. There is no required starting point.
Because we can choose the starting point for most vectors, we can slide them around on the coordinate grid to visualize problems. For example, to add vectors, it is sometimes easiest to connect the vectors “end to start”. When you connect the vectors end to start, it is called a triangle drawing.
In the figure below, a^ G = 5 i + 2 j and
G
b = − 1 i + 2 j. Their starting points are irrelevant in the first diagram. To picture a G + , slide the vectors to connect the end of vector
G
b a G to the start of vector .
G
b
G
a
G
b
Another method for showing vector addition and subtraction is the parallelogram method. Instead of drawing triangles, we slide both vectors to the same starting point; the 2 vectors form two adjacent sides of a parallelogram. When adding , the resultant is the diagonal from the starting point to the opposite corner. When subtracting , the resultant is the diagonal from the end point of the vector being subtracted back to the end point of the first vector.
Example: m^ G = − 2 i + 5 j and n G = 3 i + j are drawn from the same starting point.
Copy the vectors to form a parallelogram.
The diagonal from the starting point to the opposite corner is the resultant.
The resultant vector is i + 6 j. Algebraically, m G + n = (− 2 i + 5 j ) + (3 i + j ) = i + 6 j.
