Math 150 – Fall 2015
Section 9B
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Section 9B – Scalar Multiplication
Definition. The operation of scalar multiplication takes a scalar, that is a real
number, and multiplies a vector by it. In R2 (the plane), we define scalar multiplication
of a scalar a and a vector hx1 , x2 i as
ahx1 , x2 , i = hax1 , ax2 i.
Example 1. For the scalar multiplication of 2 times the vector h1, −2i we get
2h−1, 2i = h−2, 4i
which has the graph below
Note.
• When we multiply a vector by a scalar, if the scalar is positive, the resulting vector has the same direction as the original. If the scalar is negative, the
resulting vector has the opposite direction as the original vector.
• When we multiply a vector by a scalar, the length of the new vector is the length
of the old vector times the scalar, i.e., multiplying by the scalar 2 doubles the
length, but multiplying by the scalar 12 cuts the length in half.
Example 2. Find the scalar multiple of h−4, 7i by the scalar a = 12 . Sketch the original
vector and resulting vector.
Math 150 – Fall 2015
Section 9B
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Example 3. Find the scalar multiple of h5, 6i by the scalar a = −4. Sketch the original
vector and resulting vector.
Example 4. Does the equation ah5, −2i = h15, −6i have a solution? If it does, then
find it.
Definition. For vectors in Rn (i.e., a vector in n-dimensions with n entries), we define
scalar multiplication as
ahx1 , x2 , . . . , xn i = hax1 , ax2 , . . . , axn i
Example 5. Find the scalar multiple of h3, −7, 2i by the scalar
−1
7 .
Example 6. Find the scalar multiple of h10, −3, 8, 12i by the scalar multiple a = 5.