Vector Properties

Equal Vectors Do Not Depend on Location

Two vectors are equal if they have the same magnitude and the same direction. Their positions in a diagram may be different. This matters because forces, displacements, and velocities are often drawn at different places while still representing the same vector.

Two vectors are equal because their lengths and directions match, not because their tails sit at the same point.

Equal and Opposite Vectors

Vectors

\vec{a}

and

\vec{b}

are equal because they are parallel, point the same way, and have the same length. Vector

-\vec{a}

points the opposite way.

In the visual, $\vec{a}$ and $\vec{b}$ are not drawn on the same line, but they are still equal as vectors. Moving the drawing does not change the vector's magnitude or direction.

A Negative Vector Reverses Direction

The negative of $\vec{a}$ is written as $-\vec{a}$ . It has the same magnitude as $\vec{a}$ , but points in the opposite direction.

\lvert-\vec{a}\rvert=\lvert\vec{a}\rvert

For example, a $12\text{ N}$ force to the right and a $12\text{ N}$ force to the left form an opposite-vector pair. If both act on the same object along the same line, their resultant can be the zero vector.

Scalar Multiplication Changes Length

A scalar is an ordinary number with no direction. When a vector is multiplied by a scalar $k$ , the vector length changes by a factor of $\lvert k\rvert$ .

Form	Result magnitude	Result direction
$2\vec{a}$	twice $\vec{a}$	same as $\vec{a}$
$\frac{1}{2}\vec{a}$	half of $\vec{a}$	same as $\vec{a}$
$-3\vec{a}$	three times $\vec{a}$	opposite to $\vec{a}$
$0\vec{a}$	$0$	becomes the zero vector

A negative scalar does not make the magnitude negative. It reverses the vector direction.

The Zero Vector as Cancellation

The zero vector is written as $\vec{0}$ . Its magnitude is $0$ , and its direction is undefined because the tail and tip coincide.

The zero vector often appears when vectors cancel each other. For example, two students may pull a box with equal forces in opposite directions. If the forces are exactly along the same line, their combination does not produce a net push to the right or to the left.

\vec{F}_1+\vec{F}_2=\vec{0}

This idea becomes important in equilibrium: an object can remain at rest not because no forces act on it, but because the resultant of all forces is zero.

Equal Vectors Do Not Depend on Location

Two vectors are equal because their lengths and directions match, not because their tails sit at the same point.

Equal and Opposite Vectors

Vectors

\vec{a}

and

\vec{b}

are equal because they are parallel, point the same way, and have the same length. Vector

-\vec{a}

points the opposite way.

In the visual,

\vec{a}

and

\vec{b}

are not drawn on the same line, but they are still equal as vectors. Moving the drawing does not change the vector's magnitude or direction.

A Negative Vector Reverses Direction

The negative of

\vec{a}

is written as

-\vec{a}

. It has the same magnitude as

\vec{a}

, but points in the opposite direction.

\lvert-\vec{a}\rvert=\lvert\vec{a}\rvert

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

For example, a

12\text{ N}

force to the right and a

12\text{ N}

force to the left form an opposite-vector pair. If both act on the same object along the same line, their resultant can be the zero vector.

Scalar Multiplication Changes Length

A scalar is an ordinary number with no direction. When a vector is multiplied by a scalar

k

, the vector length changes by a factor of

\lvert k\rvert

Form	Result magnitude	Result direction
$2\vec{a}$	twice $\vec{a}$	same as $\vec{a}$
$\frac{1}{2}\vec{a}$	half of $\vec{a}$	same as $\vec{a}$
$-3\vec{a}$	three times $\vec{a}$	opposite to $\vec{a}$
$0\vec{a}$	$0$	becomes the zero vector

A negative scalar does not make the magnitude negative. It reverses the vector direction.

The Zero Vector as Cancellation

The zero vector is written as

\vec{0}

. Its magnitude is

0

, and its direction is undefined because the tail and tip coincide.

\vec{F}_1+\vec{F}_2=\vec{0}

This idea becomes important in equilibrium: an object can remain at rest not because no forces act on it, but because the resultant of all forces is zero.