Scaling & Subtracting

Stretching, flipping, unit vectors and subtraction as adding the reverse.

8 minUpdated 21 October 2026scalar multiplicationunit vectornegative vectorsubtraction

Lesson at a glanceShow the key points

k·a has magnitude |k|a; it points along a if k > 0 and opposite to a if k < 0.
A unit vector has length 1 and only carries direction: a = a·â.
a − b is defined as a + (−b).
Drawn from a common tail, a − b runs from the head of b to the head of a.
|a − b| = √(a² + b² − 2ab cos θ).

Multiply an arrow by a number and it stretches. Multiply by a negative number and it turns around.

Scaling a vector

Take a vector $\vec a$ of magnitude $a$ and an ordinary number $k$ . The product $\vec b = k\vec a$ is defined to be a vector of magnitude $|k|\,a$ :

if $k > 0$ , it points the same way as $\vec a$ ;
if $k < 0$ , it points the opposite way.

Multiplying by −1 keeps the length and reverses the direction. So $\vec a$ and $-\vec a$ are equally long and point in opposite directions.

Interactive

Multiplying a vector by a number

Slide k. Positive k stretches or shrinks the arrow along its own line; negative k also flips it around.

|\vec a|

2.24

|k\vec a|

4.47

\text{direction}

same as a

Multiplier

k

2.0

Definition · Unit vector

A vector of magnitude exactly 1, used purely to carry a direction. If

\hat u

is the unit vector along

\vec a

, then

\vec a = a\,\hat u

: the size times the direction.

This split into “how much” and “which way” comes up constantly. In the next lesson the three unit vectors $\hat\imath, \hat\jmath, \hat k$ along the coordinate axes will turn vector problems into arithmetic.

Subtracting a vector

Subtraction needs no new rule. We define $\vec a - \vec b$ as the sum of $\vec a$ and $-\vec b$ :

Subtraction

\vec a - \vec b = \vec a + (-\vec b)

Reverse

\vec b

, then add it head-to-tail.

Interactive

Subtraction is adding the reverse

Drag a and b. ‘Flip and add’ shows a + (−b). ‘Tip to tip’ shows the shortcut: a − b runs from the head of b to the head of a.

|\vec a|

4.47

|\vec b|

3.16

|\vec a - \vec b|

3.16

Picture

Because $-\vec b$ makes an angle of $180^\circ - \theta$ with $\vec a$ , the magnitude formula changes only by a sign:

|\vec a - \vec b| = \sqrt{a^2 + b^2 - 2ab\cos\theta}

Worked example

Sum and difference of equal vectors

Vectors $\vec A$ and $\vec B$ each have magnitude 8 units, and the angle between them is 60°. Find $|\vec A + \vec B|$ and $|\vec A - \vec B|$ .

What to remember

1 $k\vec a$ has length $|k|a$ ; a negative $k$ reverses the direction.
2Any vector = magnitude × unit vector: $\vec a = a\hat u$ .
3 $\vec a - \vec b$ means $\vec a + (-\vec b)$ ; drawn from a common tail it runs from the head of b to the head of a.

Check yourself

If |a| = 3, what is the magnitude of −2a?

For two vectors of equal magnitude, when is |a − b| larger than |a + b|?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadVectors (text lesson)3Blue1Brown ReadVectorsMaths is Fun ReadVectors and spacesKhan Academy

PreviousAdding Vectors Next Components & Unit Vectors

Scaling & Subtracting

Stretching, flipping, unit vectors and subtraction as adding the reverse.

8 minUpdated 21 October 2026scalar multiplicationunit vectornegative vectorsubtraction

Lesson at a glanceShow the key points

k·a has magnitude |k|a; it points along a if k > 0 and opposite to a if k < 0.
A unit vector has length 1 and only carries direction: a = a·â.
a − b is defined as a + (−b).
Drawn from a common tail, a − b runs from the head of b to the head of a.
|a − b| = √(a² + b² − 2ab cos θ).

Multiply an arrow by a number and it stretches. Multiply by a negative number and it turns around.

Scaling a vector

Take a vector $\vec a$ of magnitude $a$ and an ordinary number $k$ . The product $\vec b = k\vec a$ is defined to be a vector of magnitude $|k|\,a$ :

if $k > 0$ , it points the same way as $\vec a$ ;
if $k < 0$ , it points the opposite way.

Multiplying by −1 keeps the length and reverses the direction. So $\vec a$ and $-\vec a$ are equally long and point in opposite directions.

Interactive

Multiplying a vector by a number

Slide k. Positive k stretches or shrinks the arrow along its own line; negative k also flips it around.

|\vec a|

2.24

|k\vec a|

4.47

\text{direction}

same as a

Multiplier

k

2.0

Definition · Unit vector

A vector of magnitude exactly 1, used purely to carry a direction. If

\hat u

is the unit vector along

\vec a

, then

\vec a = a\,\hat u

: the size times the direction.

Subtracting a vector

Subtraction needs no new rule. We define $\vec a - \vec b$ as the sum of $\vec a$ and $-\vec b$ :

Subtraction

\vec a - \vec b = \vec a + (-\vec b)

Reverse

\vec b

, then add it head-to-tail.

Interactive

Subtraction is adding the reverse

Drag a and b. ‘Flip and add’ shows a + (−b). ‘Tip to tip’ shows the shortcut: a − b runs from the head of b to the head of a.

|\vec a|

4.47

|\vec b|

3.16

|\vec a - \vec b|

3.16

Picture

Because $-\vec b$ makes an angle of $180^\circ - \theta$ with $\vec a$ , the magnitude formula changes only by a sign:

|\vec a - \vec b| = \sqrt{a^2 + b^2 - 2ab\cos\theta}

Worked example

Sum and difference of equal vectors

Vectors $\vec A$ and $\vec B$ each have magnitude 8 units, and the angle between them is 60°. Find $|\vec A + \vec B|$ and $|\vec A - \vec B|$ .

What to remember

1 $k\vec a$ has length $|k|a$ ; a negative $k$ reverses the direction.
2Any vector = magnitude × unit vector: $\vec a = a\hat u$ .
3 $\vec a - \vec b$ means $\vec a + (-\vec b)$ ; drawn from a common tail it runs from the head of b to the head of a.

Check yourself

If |a| = 3, what is the magnitude of −2a?

For two vectors of equal magnitude, when is |a − b| larger than |a + b|?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadVectors (text lesson)3Blue1Brown ReadVectorsMaths is Fun ReadVectors and spacesKhan Academy

PreviousAdding Vectors Next Components & Unit Vectors

Scaling a vector#

Subtracting a vector#

Check yourself

Keep exploring

Scaling a vector#

Subtracting a vector#

Check yourself

Keep exploring

Scaling a vector

Subtracting a vector

Scaling a vector

Subtracting a vector