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Multiply a Vector by a Scalar

Related Topics:
Vectors, Scalar Multiplication of Vectors



 

A scalar quantity has a magnitude but no direction.

In vectors, a fixed numeric value is called a scalar.
A vector can be multiplied by a scalar.
When a vector x is multiplied by 3, the result is 3x.
When a vector x is multiplied by –2, the result is –2x.

In the above examples, 3 and –2 are scalars.

Diagram of scalar multiplication of a vector, positive scalar


Diagram of scalar multiplication of a vector, negative scalar

Diagram of scalar multiplication of a vector, fractional scalar




Multiplying a vector by a scalar
Multiplying a Vector by a Scalar
How to multiply a vector by a scalar including some algebraic properties of scalar multiplication.
Discuss the concept of a linear combination of vectors and shows an example of drawing a geometric sum/difference of 3 vectors.
Multiplying a vector by a scalar (real number) means taking a multiple of a vector.
Two vectors are said to be collinear when they are drawn tail to tail and they lie on the same line.
If a and b are two non-collinear vectors in the plane, then any other vector can be written as a linear combination of a and b.

Properties of Scalar Multiplication
Let v be any vector and k be a scalar. Then kv is a vector and kv is |k| times as long as v.
• If k > 0, kv has the same direction as v.
• If k < 0, kv has the opposite direction as v.
• If k = 0, kv is the zero vector.

Examples:
1. Given the vector 75 km/h [N 50° E] draw twice this vector.
2. OF = 3b and OE = 2a, write each of the following in terms of a and b.
a) EF
b) OG

How to show geometrically what multiplication of a vector by a scalar is?
Understanding multiplying vectors by scalars
Examples:
w = (1,2)
Find 3w
Find -2w


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