A sum gives a vector, a dot product gives a number: checking perpendicularity

1. Add horizontal and vertical components separately

The vector (a, b) represents a displacement of a horizontally and b vertically. A negative component means movement opposite to that axis’s positive direction. Following it with (c, d) gives total displacement (a + c, b + d). Distinguish the starting position from the displacement.

The first example asks for (2, −1) + (−3, 4). Horizontally, 2 − 3 = −1; vertically, −1 + 4 = 3. The result is (−1, 3). This displacement is unchanged if the journey starts somewhere other than the origin.

2. A dot product adds products of matching components

For plane vectors u = (a, b) and v = (c, d), define u · v = ac + bd. Addition returns a pair of components; a dot product returns one number. Check the operation symbol before calculating.

In the second example, (1, 2) · (3, −1) = 1 × 3 + 2 × (−1) = 1. Multiply horizontal by horizontal and vertical by vertical, then add. For our total displacement (2, 3) and another vector (3, −2), the dot product is 2 × 3 + 3 × (−2) = 0.

3. Why does a zero dot product mean perpendicular?

By Pythagoras, the squared length of u = (a, b) is a² + b². Draw both vectors from the same starting point. The squared distance between their tips is (a − c)² + (b − d)². Expanding gives the squared length of u plus the squared length of v minus 2(ac + bd).

For two nonzero vectors that are perpendicular, Pythagoras says that the squared distance between their tips equals the sum of their squared lengths. Comparing with the expansion gives ac + bd = 0. Conversely, a zero dot product gives that length equality, so the converse of Pythagoras establishes perpendicularity.

For (2, 3) and (3, −2), each squared length is 13. The squared distance between the tips is (2 − 3)² + (3 + 2)² = 26, which equals 13 + 13. The calculation of a zero dot product now connects to a right angle.

4. Use the three questions to distinguish displacement, number and condition

Solve the examples in order. Explain that the first asks for a displacement pair, the second for a numerical dot product, and the third for a vector satisfying perpendicularity. In the third, (2, 1) · (1, −2) = 2 × 1 + 1 × (−2) = 0. Calculate the other listed candidates too, checking that exactly one gives zero.

Return to graphs and coordinates if signs or component order are unclear. Next, use the related geometry article to reinforce reasoning from stated conditions when identifying right angles and heights.