ACT® · Math: Number and Quantity
Vector Addition and Scalar Multiplication
A vector has both size and direction, so you add or scale it one component at a time, never by combining magnitudes directly. The sum's length comes from the Pythagorean theorem applied to the new components.
A vector is an arrow: it has a length (magnitude) and a direction, and you can describe it with components like (x, y), the horizontal and vertical steps it takes. To add two vectors, add their matching components separately: (3, -1) + (-5, 4) = (-2, 3). Picture this as tip-to-tail: draw the first arrow, then start the second arrow's tail exactly where the first one's tip ended. The straight arrow from the very start to the very end is the sum, called the resultant. This is why a swimmer heading straight across a current ends up downstream: their swimming velocity and the river's current velocity add as vectors, and the resultant points diagonally even though the swimmer aimed straight across.
Scalar multiplication stretches or shrinks a vector and can flip it. Multiplying by -2 means multiply each component by -2: it doubles the length and reverses the direction, since the negative sign flips the arrow 180 degrees. Once you have final components, the magnitude comes from the Pythagorean theorem: √(x2 + y2). For a 12 newton pull east, (12, 0), combined with a 5 newton pull north, (0, 5), the resultant is (12, 5), with magnitude √(144 + 25) = 13.
This component-by-component habit is the whole topic: every vector question on the test, whether it asks for a sum, a difference, or a scaled combination like u + t times v, reduces to adding or scaling coordinates first and taking the square root last.
Key terms
- Vector
- A quantity with both magnitude and direction, often written as components (x, y).
- Magnitude
- The length of a vector, found from its components with √(x2 + y2).
- Resultant
- The single vector produced by adding two or more vectors together.
- Scalar multiplication
- Multiplying every component of a vector by the same number, which scales and can reverse its direction.
- Tip-to-tail
- A method of adding vectors by placing the tail of one arrow at the tip of another, then drawing the sum from start to end.
Watch it explained
- How it worksHow does parking one arrow's tail on the other arrow's tip build their sum?
- How it worksHow does multiplying a vector by -2 reverse it and stretch it in the same stroke?
- Worked exampleFind the sum of the vectors (3, -1) and (-5, 4) and draw the tip-to-tail picture.
- Worked exampleFind the resultant of a 12 newton pull due east and a 5 newton pull due north.
- Common mistakeWhy isn't subtracting two vectors the same as subtracting their magnitudes?
Check yourself
End of topic.