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Magnitude & Direction
of Scaled Vectors

Learn how scalar multiplication changes a vector's length and direction: the key formula \(\|c\mathbf{v}\| = |c|\,\|\mathbf{v}\|\) and the rule for when \(c\mathbf{v}\) points along or against \(\mathbf{v}\).

Precalculus

What Is a Vector?

A vector is a quantity that has both magnitude (length) and direction. We write a vector as \(\mathbf{v}\) and its magnitude as \(\|\mathbf{v}\|\). In component form: \(\mathbf{v} = \langle v_1, v_2 \rangle\).

x y
\(\mathbf{v}\)
\(v_1\)
\(v_2\)
 O

What Is Scalar Multiplication?

When we multiply a vector \(\mathbf{v} = \langle v_1, v_2 \rangle\) by a real number (scalar) \(c\), we get:

$$c\mathbf{v} = \langle c\,v_1,\; c\,v_2 \rangle$$

Stretches when \(|c| > 1\)
The vector gets longer: every component is multiplied by more than 1.
Shrinks when \(0 < |c| < 1\)
The vector gets shorter: every component is multiplied by a fraction.
Reverses when \(c < 0\)
The vector flips direction (points opposite) and scales by \(|c|\).

Visualizing Scalar Multiples

O
\(\mathbf{v}\)
\(2\mathbf{v}\)
\(0.5\mathbf{v}\)
\(-\mathbf{v}\)
\(-2\mathbf{v}\)

The Magnitude Rule

$$\|c\mathbf{v}\| = |c|\;\|\mathbf{v}\|$$

The magnitude of a scaled vector equals the absolute value of the scalar times the original magnitude. This means:

1
Magnitude is always non-negative. Even if \(c\) is negative, the absolute value ensures \(\|c\mathbf{v}\| \geq 0\).
2
The sign of \(c\) does not affect length. Both \(3\mathbf{v}\) and \(-3\mathbf{v}\) have the same magnitude: \(3\|\mathbf{v}\|\).
3
Zero scalar means zero vector. If \(c = 0\), then \(\|0\mathbf{v}\| = 0\).

Why Does \(\|c\mathbf{v}\| = |c|\,\|\mathbf{v}\|\) Work?

Let \(\mathbf{v} = \langle v_1, v_2 \rangle\). Then \(c\mathbf{v} = \langle cv_1, cv_2 \rangle\). Compute the magnitude:

$$\|c\mathbf{v}\| = \sqrt{(cv_1)^2 + (cv_2)^2} = \sqrt{c^2\,v_1^2 + c^2\,v_2^2}$$

$$= \sqrt{c^2(v_1^2 + v_2^2)} = \sqrt{c^2}\;\sqrt{v_1^2 + v_2^2} = |c|\;\|\mathbf{v}\|$$

The Direction Rule

When \(|c|\,\|\mathbf{v}\| \neq 0\):

\(c > 0\): Same Direction
The vector \(c\mathbf{v}\) points along \(\mathbf{v}\). The direction angle is unchanged.
\(c < 0\): Opposite Direction
The vector \(c\mathbf{v}\) points against \(\mathbf{v}\). The direction angle shifts by \(180^\circ\).
\(c > 0\): same direction
\(\mathbf{v}\)
\(2\mathbf{v}\)
\(c < 0\): opposite direction
\(\mathbf{v}\)
\(-\mathbf{v}\)

Magnitude vs. Direction

Magnitude \(\|c\mathbf{v}\|\) Direction of \(c\mathbf{v}\)
Depends on \(|c|\) and \(\|\mathbf{v}\|\) Sign of \(c\)
Formula \(|c|\,\|\mathbf{v}\|\) Along \(\mathbf{v}\) or against \(\mathbf{v}\)
\(c = 3\) \(3\|\mathbf{v}\|\) Same as \(\mathbf{v}\)
\(c = -3\) \(3\|\mathbf{v}\|\) Opposite to \(\mathbf{v}\)
\(c = 0\) \(0\) Undefined (zero vector)

How to Apply the Rules

1
Identify \(c\) and \(\mathbf{v}\). Write down the scalar and the vector (or its magnitude if given).
2
Compute magnitude: Take \(|c|\) (absolute value of the scalar) and multiply by \(\|\mathbf{v}\|\).
3
Determine direction: If \(c > 0\), same direction as \(\mathbf{v}\). If \(c < 0\), opposite direction.
4
Check the edge case: If \(c = 0\) or \(\mathbf{v} = \mathbf{0}\), the result is the zero vector (no direction).

Example 1: Positive Scalar

Let \(\mathbf{v} = \langle 3, 4 \rangle\) and \(c = 2\). Find \(\|c\mathbf{v}\|\) and the direction of \(c\mathbf{v}\).

1
Find \(\|\mathbf{v}\|\): \(\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
2
Compute \(\|c\mathbf{v}\|\): \(\|2\mathbf{v}\| = |2| \cdot 5 = 2 \cdot 5 = \) 10
3
Direction: Since \(c = 2 > 0\), the direction of \(2\mathbf{v}\) is the same as \(\mathbf{v}\).

Example 2: Negative Scalar

Let \(\mathbf{v} = \langle 3, 4 \rangle\) (so \(\|\mathbf{v}\| = 5\)) and \(c = -3\). Find \(\|c\mathbf{v}\|\) and its direction.

1
Absolute value: \(|c| = |-3| = 3\)
2
Magnitude: \(\|-3\mathbf{v}\| = 3 \cdot 5 = \) 15
3
Direction: Since \(c = -3 < 0\), the direction of \(-3\mathbf{v}\) is opposite to \(\mathbf{v}\).

Example 3: Fractional Scalar

A vector \(\mathbf{u}\) has magnitude \(\|\mathbf{u}\| = 12\) and points northeast. Find the magnitude and direction of \(-\frac{1}{4}\mathbf{u}\).

1
Scalar: \(c = -\frac{1}{4}\), so \(|c| = \frac{1}{4}\)
2
Magnitude: \(\left\|-\frac{1}{4}\mathbf{u}\right\| = \frac{1}{4} \cdot 12 = \) 3
3
Direction: Since \(c < 0\), the direction is opposite to \(\mathbf{u}\), i.e., southwest.

Seeing It on the Coordinate Plane

Given \(\mathbf{v} = \langle 2, 3 \rangle\), here are \(\mathbf{v}\), \(2\mathbf{v}\), and \(-1.5\mathbf{v}\):

x y
\(\mathbf{v}\)
\(2\mathbf{v}\)
\(-1.5\mathbf{v}\)

Common Mistakes to Avoid

✕ Forgetting Absolute Value
Writing \(\|c\mathbf{v}\| = c\|\mathbf{v}\|\) instead of \(\|c\mathbf{v}\| = |c|\,\|\mathbf{v}\|\). When \(c = -3\), this gives a negative magnitude, which is impossible.
✕ Confusing Magnitude with Direction
Thinking a negative scalar makes magnitude negative. In truth, the sign only affects direction, never length.
✕ Ignoring the Zero Case
Claiming \(0\mathbf{v}\) has a direction "along \(\mathbf{v}\)." The zero vector has no defined direction.

Summary of Scalar Multiplication Effects

Scalar \(c\) Magnitude \(\|c\mathbf{v}\|\) Direction Effect
\(c > 1\) \(|c|\,\|\mathbf{v}\|\) Same as \(\mathbf{v}\) Stretch
\(c = 1\) \(\|\mathbf{v}\|\) Same as \(\mathbf{v}\) No change
\(0 < c < 1\) \(|c|\,\|\mathbf{v}\|\) Same as \(\mathbf{v}\) Shrink
\(c = 0\) \(0\) Undefined Zero vector
\(-1 < c < 0\) \(|c|\,\|\mathbf{v}\|\) Opposite to \(\mathbf{v}\) Shrink + reverse
\(c = -1\) \(\|\mathbf{v}\|\) Opposite to \(\mathbf{v}\) Reverse
\(c < -1\) \(|c|\,\|\mathbf{v}\|\) Opposite to \(\mathbf{v}\) Stretch + reverse

Question 1 of 5

If \(\|\mathbf{v}\| = 7\), what is \(\|-4\mathbf{v}\|\)?

Question 2 of 5

A vector \(\mathbf{w}\) points due east. In which direction does \(-5\mathbf{w}\) point?

Question 3 of 5

If \(\mathbf{v} = \langle 6, -8 \rangle\), what is \(\left\|\frac{1}{2}\mathbf{v}\right\|\)?

Question 4 of 5

Which statement is always true about \(c\mathbf{v}\) when \(c \neq 0\) and \(\mathbf{v} \neq \mathbf{0}\)?

Question 5 of 5

A vector \(\mathbf{u}\) has magnitude \(10\) and direction \(45^\circ\). What are the magnitude and direction of \(-2\mathbf{u}\)?