Multiplying a vector \(\vec{a}\) of magnitude \(a\) and with a positive number \(k\) gives a vector whose magnitude is changed by the factor \(k\) but the direction is the same as that of \(\vec{a}\). We define the vector \(\vec{b}=k \vec{a}\) as a vector of magnitude \(|k a|\). If \(k\) is positive the direction of the vector \(\vec{b}=k \vec{a}\) is same as that of \(\vec{a}\). If \(k\) is negative, the direction of \(\vec{b}\) is opposite to \(\vec{a}\). In particular, multiplication by \((-1\) ) just inverts the direction of the vector. The vectors \(\vec{a}\) and \(-\vec{a}\) have equal magnitudes but opposite directions. If \(\vec{a}\) is a vector of magnitude \(a\) and \(\vec{v}\) is a vector of unit magnitude in the direction of \(\vec{a}\), we can write \(\vec{a}=a \vec{v}\).

For example, if \(\mathbf{A}\) is multiplied by 2 , the resultant vector \(2 \mathbf{A}\) is in the same direction as \(\mathbf{A}\) and has a magnitude twice of \(|\mathbf{A}|\) as shown in Flg. 4.3(a).
Multiplying a vector \(\mathbf{A}\) by a negative number \(-\lambda\) gives another vector whose direction is opposite to the direction of \(\mathbf{A}\) and whose magnitude is \(\lambda\) times \(|\mathbf{A}|\).

Multiplying a given vector A by negative numbers, say \(-1\) and \(-1.5\), gives vectors as shown in Fig 4.3(b).

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