J: Vectors: Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.

The idea

Vectors can be added by placing them tip to tail: the resultant is the arrow from the start of the first vector to the tip of the last. This is the triangle law, extended to the parallelogram law when two vectors share a starting point. Algebraically, add corresponding components: (a, b) + (c, d) = (a + c, b + d). Multiplying by a scalar k stretches the vector by factor k: k(a, b) = (ka, kb). If k is negative, the direction reverses; if 0 < k < 1, the vector shrinks. Geometrically, scalar multiples are parallel vectors, and a linear combination such as 2a - 3b can be constructed by drawing scaled copies and adding tip to tail. For example, if a = (2, 1) and b = (-1, 3), then 2a - 3b = (4, 2) + (3, -9) = (7, -7). These operations underpin position vectors, midpoints and proofs of collinearity.

What earns the marks

• Draws vectors tip to tail accurately and labels the resultant with a double arrow or clear direction.
• Adds component-wise correctly, including negative components, and simplifies to a single column vector.
• Multiplies each component by the scalar, handling negative and fractional scalars correctly.
• Interprets scalar multiples geometrically as parallel vectors, noting reversal for negative scalars.
• Uses vector addition and scalar multiplication in context, such as finding a position vector or proving points are collinear.

Common mistakes and how to put them right

• Adding vectors by adding their magnitudes rather than their components.

  Put it right: Add corresponding components; magnitudes only add when vectors are parallel and in the same direction.

• Drawing the resultant from tip to tip instead of from the start of the first vector to the tip of the last.

  Put it right: The resultant closes the triangle from start to finish.

• Sign errors when subtracting vectors.

  Put it right: Subtract each component of b, or add the negative of b, which reverses its direction.

Exam tips

• Use column vectors for algebraic work and diagrams for checking; the two methods should agree.
• When a question says 'show that', set out each vector step clearly so the reasoning is visible, not just the final answer.
• For collinearity, show one vector is a scalar multiple of another and state that they share a common point.