Vectors

Vectors are a way of expressing magnitude (size) and direction. The magnitude of a vector will often be described using variables, most commonly the variables i and j. The direction of the vector will be indicated by an arrowhead at the end of the vector line.

Vector Skills

• There are three main operations students might be asked to perform on vectors: addition, subtraction, and multiplication.

Adding

To add together any two vectors, take each vector and attach it to the endpoint of the vector that it is being added to. This will create a diamond shape. The diamond starts where the two vectors start, and it ends where they meet (after being added on to the other vector).

The sum of the two vectors is then equal to a new vector created by going from one end of the diamond to the other. The direction/arrow of this new vector will move in the same direction as the two added vectors.

Subtracting

Subtracting vectors is similar to addition. However, instead of simply attaching both vectors to their ends of their added pairs, first one of the vectors has to be flipped.

Specifically, the vector that is being subtracted needs to have it’s direction flipped (though it’s magnitude remains the same).

This flipped vector is then added to the other, original vector using the same diamond pattern from above. The product of the subtraction is once again equal to the vector created between the two ends of the diamond.

Scaling Vectors (Scalar Multiplication)

On the ACT, the only kind of vector multiplication that a student might be asked to know is scalar multiplication. Like the name suggests, this is simply scaling the vector up or down (either growing or shrinking it). There are ways to multiply actual vectors together, but this should not be covered on the ACT Math section.

Scalar Multiplication follows some quick and simple rules:

• If a vector is multiplied by a number whose magnitude is greater than 1, then the vector will increase in magnitude (length) by that number
• If a vector is multiplied by a number whose magnitude is less than 1, then the vector will shrink in magnitude (length) by that number
• If a vector is multiplied by a positive number, then the direction of vector will not change
• If a vector is multiplied by a negative number, then the direction of vector will be reversed

i and j notation

Vector problems sometimes use the variables iand j. These are known as the unit vectors, and each variable is associated with a specific axis. i is the unit vector for the x-axis. j is the unit vector for the y-axis.

When these show up on an ACT math question, they will often be used to refer to direction. For example, here is a sample question on vectors from an ACT Diagnostic test:

Since vector j represents 1 mile per hour north, that means that  − j would represent 1 mile per hour south. Maria is jogging south at 12 miles per hour, meaning that  − j would need to be scaled up by 12 to properly represent her velocity.

Therefore, the answer is 12 • − j =  − 12j or B.