J: Vectors: Use vectors in two dimensions and in three dimensions.

The idea

A vector has both magnitude and direction. In 2D you may write a vector as a column vector or in i, j notation; in 3D you extend this to i, j, k. You must add, subtract and multiply by scalars component-wise, and find magnitude using Pythagoras in 2D or its 3D extension. For example, if a = 3i - 2j + k and b = i + 4j - 2k, then a + b = 4i + 2j - k and |a| = sqrt(3^2 + (-2)^2 + 1^2) = sqrt(14). You also use vectors to solve geometric problems: the vector from A to B is AB = b - a, the midpoint of AB has position vector (a + b)/2, and two vectors are parallel when one is a non-zero scalar multiple of the other, which also tests collinearity. In context, vectors may represent forces, velocities or displacements, so interpret results with magnitude and direction.

What earns the marks

• Add, subtract and multiply vectors by scalars component-wise in both 2D and 3D, keeping i, j and k terms separate.
• Calculate magnitude using |v| = sqrt(x^2 + y^2) in 2D and |v| = sqrt(x^2 + y^2 + z^2) in 3D, showing the squares and the square root.
• Use position vectors to find the vector between two points, AB = b - a, and the midpoint, (a + b)/2, in geometric problems.
• Show two vectors are parallel by writing one as a scalar multiple of the other, e.g. b = k a, and use this to test whether three points are collinear.
• Interpret a vector result in context, giving magnitude with correct units and a direction where required, such as a velocity as a speed and bearing.

Common mistakes and how to put them right

• Adding magnitudes instead of components, e.g. treating |a| + |b| as |a + b|.

  Put it right: Add corresponding components separately, then find the magnitude of the resultant if needed.

• Forgetting to square each component or omitting the square root when finding magnitude.

  Put it right: Use |v| = sqrt(x^2 + y^2) in 2D and |v| = sqrt(x^2 + y^2 + z^2) in 3D.

• Assuming two vectors are parallel because they have the same magnitude.

  Put it right: Check that one is a scalar multiple of the other, so the components are in the same ratio.

Exam tips

• Write vectors clearly using underlined letters or column notation, and show each component calculation so accuracy marks are easy to award.
• Draw a diagram for geometric problems, labelling position vectors, and set up equations such as AB = b - a before calculating.