8.2 Multiplication of Vector by a Scalar and the Parallel Condition of Two Vectors

8.2 Multiplication of Vector by a Scalar and the Parallel Condition of Two Vectors

1. When a vector

\underset{˜}{a}

is multiplied by a scalar k, the product is

k \underset{˜}{a}

. Its magnitude is k times the magnitude of the vector

\underset{˜}{a}

2. The vector

\underset{˜}{a}

is parallel to the vector

\underset{˜}{b}

if and only if

\underset{˜}{b} = k \underset{˜}{a}

, where k is a constant.

3. If the vectors

\underset{˜}{a}

and

\underset{˜}{b}

are not parallel and

h \underset{˜}{a} = k \underset{˜}{b}

, then h = 0 and k = 0.

Example 1:

If vectors

\underset{˜}{a} and \underset{˜}{b}

are not parallel and

(k - 7) \underset{˜}{a} = (5 + h) \underset{˜}{b}

, find the value of k and of h.

Solution:

The vectors

\underset{˜}{a} and \underset{˜}{b}

are not parallel, so

k – 7 = 0 → k = 7

5 + h = 0 → h = –5