School of Mathematics and Statistics, The University of Sydney
 7. Cartesian coordinates in three dimensions
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Multiplication by scalars in terms of components

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If a vector is expressed in Cartesian form, then it’s easy to calculate any scalar multiple of that vector in Cartesian form.

The general rule is that given any vector v = xi + yj + zk and any scalar a, then

av = axi + ayj + azk.

For example, if v = 2i + j - k, then -4v = -8i - 4j + 4k.

The same formula applies in two dimensions. The vector v = xi + yj can be thought of as v = xi + yj + 0k. Then if a is any scalar,

av = axi + ayj + (a × 0)k = axi + ayj + 0k = axi + ayj.

Thus if u = 2i + j then 2u = 4i + 2j, as illustrated in the following picture.

2j 2u S u j Q R T O 2i 4i

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