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CIE A-Level Maths Study Notes

2.7.2 Vector Operations and Geometric Interpretations

Vectors are essential in mathematics for representing quantities with both magnitude and direction. This section explores vector operations and their applications in geometry, enabling a deeper understanding of mathematical concepts and physical phenomena.

Vector Operations

Addition and Subtraction

Addition and Subtraction of Vectors

Image courtesy of blogspot

Combine vectors by adding or subtracting corresponding components.

  • Addition Example: a+b=(3i+2j)+(4ij)=7i+j\mathbf{a} + \mathbf{b} = (3\mathbf{i} + 2\mathbf{j}) + (4\mathbf{i} - \mathbf{j}) = 7\mathbf{i} + \mathbf{j}
  • Subtraction Example: ab=(3i+2j)(4ij)=i+3j\mathbf{a} - \mathbf{b} = (3\mathbf{i} + 2\mathbf{j}) - (4\mathbf{i} - \mathbf{j}) = -\mathbf{i} + 3\mathbf{j}

Multiplication by a Scalar

Alter the magnitude of a vector by multiplying it with a scalar.

  • Scalar Multiplication Example: 2a=2(3i+2j)=6i+4j2\mathbf{a} = 2(3\mathbf{i} + 2\mathbf{j}) = 6\mathbf{i} + 4\mathbf{j}

Magnitude and Normalisation

  • Magnitude Example:
    • For a=3i+4j),(a=32+42=5\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}), (|\mathbf{a}| = \sqrt{3^2 + 4^2} = 5.
  • Unit Vector Example:
    • For a=3i+4j),(a^=15(3i+4j)=0.6i+0.8j\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}), (\hat{\mathbf{a}} = \frac{1}{5}(3\mathbf{i} + 4\mathbf{j}) = 0.6\mathbf{i} + 0.8\mathbf{j}.

Geometric Interpretations

Displacement Vector

Represents the shortest route between two points.

Example:

If point A is at (2, 3) and point B is at (5, 7), then AB=(52)i+(73)j=3i+4j\vec{AB} = (5-2)\mathbf{i} + (7-3)\mathbf{j} = 3\mathbf{i} + 4\mathbf{j}.

Position Vector

Defines a point's position in relation to the origin.

Example:

If point A is at (2, 3), then OA=2i+3j\vec{OA} = 2\mathbf{i} + 3\mathbf{j}.

Geometrical Principles

Parallelogram Law

Example:

In parallelogram OABC, if OA=3i+2j\vec{OA} = 3\mathbf{i} + 2\mathbf{j} and OC=i+5j\vec{OC} = \mathbf{i} + 5\mathbf{j}, then OB=OA+OC=4i+7j\vec{OB} = \vec{OA} + \vec{OC} = 4\mathbf{i} + 7\mathbf{j}.

Midpoint of a Vector Segment

Example:

For a segment AB\vec{AB} with endpoints A(2, 3) and B(8, 11), the midpoint M is found by OM=12[(2i+3j)+(8i+11j)]=5i+7j\vec{OM} = \frac{1}{2}[(2\mathbf{i} + 3\mathbf{j}) + (8\mathbf{i} + 11\mathbf{j})] = 5\mathbf{i} + 7\mathbf{j}.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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