What Is A Vector? (Definition, Multiplication & Examples)

Written by

Malcolm McKinsey

January 12, 2023

Fact-checked by

Paul Mazzola

Vector definition

A vector is a quantity in mathematics that has a magnitude (distance, velocity, or size) and a direction (as on a compass needle, like west, up, southeast, down, or north by northwest).

Rowing a boat across an inlet, you may think you are rowing due south at 3 knots, but if the tides are receding, you could be moving at 5 knots southeast.

A vector or several vectors working together will take into account the distance you row, your speed, and your actual direction.

Vector symbol

To represent vectors, mathematicians, physicists, and engineers use rays, labeling them with lowercase or uppercase letters, like this:

Vector addition and subtraction

Simple vector math is not too complicated.

To add vectors, we connect the tail of one vector to the head of another, using an arrow. The direct-line ray connecting the two vectors is the resultant, r, as in this drawing:

We add vector CD to vector EF and get resultant r.

Here's a stumper: does this figure give the same results as the previous figure?

Here we add vector EF to vector CD and still get resultant r. Vectors follow the same rules of arithmetic as whole numbers (here, the commutative property).

In the real world, taking the two trips along the vectors may show you entirely different scenery. In mathematics, since the two vectors did not change their direction or magnitude, the resultants are identical: CD + EF = EF + CD = r.

Vector subtraction only takes two steps:

1. negate the vector you wish to subtract

2. then add the two vectors together

Here are vectors SA and IL, showing the route we thought we were taking on a sailboat:

We did not realize the current was strong; we got lost in a fog; the sun was blinding. For whatever reason, instead of following vector IL, we went in the opposite direction. So instead of adding vector IL, we need to subtract it. We do that by negating vector IL and adding it to vector SA:

Multiplying vectors by a scalar

Multiplying a vector by a scalar (a real number) is called scalar multiplication.

Vectors have two parts (magnitude and direction), but we cannot multiply direction. That makes no sense: two "souths" are not somehow more south-facing than one "south." But we can multiply the vector's magnitude:

• Multiplying a vector times a positive whole-number scalar > 1 produces a larger vector.

• Multiplying a vector times a negative whole-number scalar < −1 produces a larger vector in the opposite direction.

• Multiplying a vectors times 1 returns the same vector (0 displacement).

• Multiplying a vector times a positive fractional scalar < 1 produces a smaller vector.

• Multiplying a vector times a negative fractional scalar > −1 produces a smaller vector in the opposite direction.

Two vectors can also be multiplied by each other using the cross product or dot product.

Cross product multiplication of two vectors produces a new vector, and the dot product produces a number, sometimes known as the scalar product.

Magnitude of a vector

The magnitude of a vector is shown as an absolute value, |a|, or with two lines to avoid confusing it with absolute value, ||a||.

If you know the x-axis and y-axis values of a vector (as if it were on a map or cartesian coordinate system) you can easily calculate its magnitude by applying the Pythagorean Theorem to the change in position from tail to arrowhead:

So here, with the tail at (1, 4) and the arrowhead at (7, 8), we have a change in x-value of 6 and a change in y-value of 4, so:

The magnitude of the vector is 7.2111 units.

The unit is determined by what you are measuring; inches, kilometers, miles per hour (mph), etc. So, if we just measured distance in miles, then 7.2111 miles would be the length of the vector.

Scalar vs. vectors

To be clear, scalar quantities are magnitude only: mass, temperature, speed, volume, distance, energy, work, and so forth. Think of them as pure numbers.

Displacement vectors

Flying superheroes seldom take the shortest route from Daily Bugle or Daily Planet to that day's disaster. They swoop, loop, leap and barrel roll before finally arriving in the nick of time.

If we used vectors to chart a flying superhero's course, we might need five or six vectors to take into account all those detours. A displacement vector cuts from start to finish in a straight line:

Displacement in this meaning comes from physics, meaning the change in position from the initial position.

You put your right hand in; you take your right hand out: Zero displacement. You cha-cha three steps to the left and two steps to the right: A displacement vector of one step left.

In this figure, we can see the displacement vector is also the resultant.

The calculation of displacement is still vector n + vector v = r, because vectors a and y negate each other!

You may think we wasted a lot of effort to move so small a distance, but what if we were in a Navy vessel and had to navigate around a jetty or protected wildlife preserve? Then we can see this really was the shortest route

Vector examples

All of these measurements are examples of vectors because they all involve distance or size of a force and a direction:

• Velocity

• Force

• Acceleration

• Momentum

• Displacement

Commercial airliners, fighter jets, boats, cars, bicyclists, runners, falling objects, rockets, hot air balloons, paper airplanes, and submarines are all examples of moving objects that use vectors in everyday life.

Pilots and navigators must use vectors to get to their destinations. Rocket scientists and aerospace engineers use vectors to control rockets.

There is one exception to vectors having length and direction, and that is the zero vector. The zero vector has no length, so it is not pointing in any particular direction. This means the zero vector has an undefined direction.

Basic vector problems

1. What will happen if we multiply a vector by 4? We hope you said it will point in the same direction but will be four times longer!

2. Is "25 knots south by southwest" a scalar or a vector? That is a vector, since it gives a magnitude and direction.

3. What will happen to a vector if we multiply it by $-\frac{1}{2}$? We hope you said it would be half as long and going in the opposite direction!

4. Two vectors are parallel but pointing in opposite directions. One is vector z. What is the other vector? We hope you remembered about negating vectors, by labeling it -z