Mathematics · Ch 10 — Vector Algebra

Multiplication of a Vector by a Scalar

10.5

Multiplication of a Vector by a Scalar

Concept: Scaling a Vector

Multiplying a vector by a number (a scalar) scales it: the vector's direction either stays the same or flips, and its length changes by a factor equal to the absolute value of that number.

Let $\vec{a}$ be any vector and $\lambda$ a real scalar. The product $\lambda \vec{a}$ is a new vector that is collinear with $\vec{a}$ , with:

Direction: if $\lambda > 0$ , $\lambda \vec{a}$ points in the same direction as $\vec{a}$ ; if $\lambda < 0$ , in the opposite direction.
Magnitude: $|\lambda \vec{a}| = |\lambda| \, |\vec{a}|$ .

Note

The scalar $\lambda$ can be any real number — positive, negative, zero, or a fraction. The vector $\lambda \vec{a}$ is always collinear with $\vec{a}$ .

Special Cases of Scalar Multiplication

The Zero Scalar

If $k = 0$ , then $k \vec{a} = \vec{0}$ for any vector $\vec{a}$ — the zero vector, with zero magnitude and undefined direction.

The Scalar $\lambda = -1$

When $\lambda = -1$ , we get the negative (or additive inverse) of $\vec{a}$ :

(-1)\vec{a} = -\vec{a}

This vector has the same magnitude as $\vec{a}$ ( $|-\vec{a}| = |\vec{a}|$ ) but points in the exactly opposite direction, and:

\vec{a} + (-\vec{a}) = (-\vec{a}) + \vec{a} = \vec{0}

Important

The negative of a vector is its additive inverse. Adding a vector and its negative always yields the zero vector.

The Scalar $\lambda = \frac{1}{|\vec{a}|}$

Provided $\vec{a} \neq \vec{0}$ , take $\lambda = \frac{1}{|\vec{a}|}$ . The magnitude of the resulting vector is:

$\left| \frac{1}{|\vec{a}|} \vec{a} \right| = \frac{1}{|\vec{a}|} \cdot |\vec{a}| = 1$ …

Figure 10.12Scalar multiplication of a vector: vector a alongside its multiples (1/2)a, 2a, -(1/2)a and -2a, showing how length scales and direction reverses as the scalar lambda changes.

Fig. 10.12 — Scalar multiplication of a vector: vector a alongside its multiples (1/2)a, 2a, -(1/2)a and -2a, showing how length scales and direction reverses as the scalar lambda changes.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your NCERT textbook's own diagram.

Fig 10.12 is a simple but powerful geometric picture. It shows a single vector a drawn as an arrow pointing up and to the right. Around it, on the same line through the origin, are four other arrows: ½a (half the length of a, same direction), 2a (twice the length, same direction), −½a (half the length but pointing opposite to a), and −2a (twice the length, opposite direction). The figure has no axes or grid — it is just a straight line with these five arrows placed along it, all collinear.

The core idea is that multiplying a vector by a scalar $\lambda$ stretches or shrinks the vector’s length by $|\lambda|$ , and flips its direction if $\lambda$ is negative. The vector $\lambda \mathbf{a}$ always lies on the same line as $\mathbf{a}$ — it is collinear with $\mathbf{a}$ . The figure makes this visually obvious: all five arrows lie on one straight line, with the positive multiples pointing the same way as $\mathbf{a}$ and the negative multiples pointing exactly opposite.

The textbook uses this figure to develop two key formulas. First, the magnitude relation:

|\lambda \mathbf{a}| = |\lambda| \, |\mathbf{a}|

Here $|\lambda \mathbf{a}|$ is the length of the scaled vector, $|\lambda|$ is the absolute value of the scalar (so length is always non-negative), and $|\mathbf{a}|$ is the original length. In the figure, $|\mathbf{a}|$ is some fixed length; $|2\mathbf{a}|$ is twice that, $|\tfrac12\mathbf{a}|$ is half, and $|{-2}\mathbf{a}|$ is also twice — the minus sign only affects direction, not magnitude.

Second, the figure leads directly to the definition of a unit vector in the direction of $\mathbf{a}$ :

\hat{\mathbf{a}} = \frac{1}{|\mathbf{a}|} \mathbf{a}, \quad \mathbf{a} \neq \mathbf{0}

The symbol $\hat{\mathbf{a}}$ (read “a-hat”) is a vector of length 1 pointing exactly along $\mathbf{a}$ . The formula says: take $\mathbf{a}$ , multiply it by the scalar $\lambda = 1/|\mathbf{a}|$ , and you get a vector of unit length. In the figure, if you imagine shrinking $\mathbf{a}$ down to a length of 1 (keeping its direction), that arrow would be $\hat{\mathbf{a}}$ . The textbook also notes that when $\lambda = -1$ , the vector $-\mathbf{a}$ is the negative or additive inverse of $\mathbf{a}$ , satisfying $\mathbf{a} + (-\mathbf{a}) = \mathbf{0}$ . …

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Multiplication of a Vector by a Scalar