Add, subtract and scalar-multiply vectors and matrices, find a vector's magnitude, and multiply small matrices (Number and Quantity).

What this topic is asking

The ACT includes a small number of vector and matrix questions in the Number and Quantity area. They test basic operations only: adding and scaling vectors, a vector's magnitude, and adding, scaling and multiplying small matrices. None require deep linear algebra, so a handful of mechanical rules covers what appears.

Vector operations

A vector lists components, usually written $\langle a, b \rangle$ or as a column.

Geometrically, adding vectors places them tip to tail, but on the ACT the component arithmetic is all you need.

Magnitude is the Pythagorean theorem

The magnitude of $\langle a, b \rangle$ is the length of the arrow from the origin to the point $(a, b)$, which by the Pythagorean theorem is $\sqrt{a^{2} + b^{2}}$. For $\langle 6, 8 \rangle$ this is $\sqrt{36 + 64} = \sqrt{100} = 10$. Recognising the common right-triangle triples (3-4-5, 6-8-10, 5-12-13) lets you read many magnitudes instantly.

Matrix operations

A matrix is a rectangular array; its size is rows by columns.

Addition and subtraction require matching sizes; you cannot add a 2-by-2 matrix to a 2-by-3 matrix.

Matrix multiplication

Multiplying matrices is the one operation with a twist. To find the entry in row $i$, column $j$ of the product, take row $i$ of the first matrix and column $j$ of the second, multiply matching entries, and add. For two 2-by-2 matrices:

The inner dimensions must match (the first matrix's column count equals the second's row count), and the product has the outer dimensions. Matrix multiplication is not commutative: $AB$ usually does not equal $BA$.

Why these stay simple on the ACT

ACT vector and matrix questions reward knowing the mechanical rules, not theory. The two reliable checks are to operate component by component (or entry by entry) for sums and scalar multiples, and to remember that magnitude is a square root of a sum of squares. For the rare matrix-multiplication item, work one entry at a time using "row times column", and confirm the sizes are compatible before you start.

Try this

Q1. If $\vec{w} = \langle -2, 5 \rangle$, what is $3\vec{w}$? [1 point]

• Cue. Multiply each component by 3: $\langle -6, 15 \rangle$.

Q2. Find the magnitude of $\langle 5, 12 \rangle$. [1 point]

• Cue. $\sqrt{25 + 144} = \sqrt{169} = 13$ (a 5-12-13 triple).