Vectors: apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors

The idea

A vector has both magnitude and direction. In column form, a vector from A to B is written as a column pair: top number is horizontal change, bottom is vertical change. For example, AB = (3, -2) means 3 right and 2 down. To add vectors, add corresponding components: (3, -2) + (1, 4) = (4, 2). To subtract, subtract corresponding components: (3, -2) - (1, 4) = (2, -6). Multiplying by a scalar multiplies each component: 2(3, -2) = (6, -4). Diagrammatically, draw vectors end-to-end for addition; reverse the second vector for subtraction. Column vectors and diagrams represent the same displacement. Always check direction signs: negative top means left, negative bottom means down.

What earns the marks

• Adds vectors by adding corresponding components, e.g. (a, b) + (c, d) = (a + c, b + d).
• Subtracts vectors by subtracting corresponding components, e.g. (a, b) - (c, d) = (a - c, b - d).
• Multiplies a vector by a scalar by multiplying each component by that scalar, e.g. k(a, b) = (ka, kb).
• Interprets a column vector as a displacement: top component horizontal, bottom component vertical, with signs indicating direction.
• Represents vector addition or subtraction diagrammatically by placing vectors end-to-end or reversing direction as appropriate.

Common mistakes and how to put them right

• Adding only the top components and ignoring the bottom components.

  Put it right by adding both corresponding components.

• Subtracting vectors in the wrong order, e.g. computing (c, d) - (a, b) instead of (a, b) - (c, d).

  Put it right by keeping the order given.

• Multiplying only one component by the scalar.

  Put it right by multiplying both components.

• Confusing the direction of a negative component, e.g. treating a negative top component as rightward.

  Put it right by interpreting negative top as left and negative bottom as down.

Exam tips

• Write column vectors clearly with brackets and separate top and bottom numbers to avoid misreading.
• When adding or subtracting, work component by component and check signs carefully.
• For diagrammatic questions, draw arrows to show direction and label vectors to avoid ambiguity.