10.1.1

Vectors - Magnitude & Direction

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Vector components

A very useful way of writing a vector is in terms of its xx and yy-components.

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Component form of a vector

  • In the xyxy-coordinate system, a point on a plane is described by a pair of coordinates (x,y)(x, y).
  • Similarly, a two dimensional vector, a\mathbf a is described by a pair of its vector coordinates:
    • a=aii+ajj\mathbf{a} = a_i\mathbf{i} + a_j\mathbf{j}
  • Where i\mathbf{i} and j\mathbf{j} are the unit vectors in the positive xx and yy directions respectively.
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Column vector

  • The equivalent vector written in column form is:
    • a=(aiaj)\mathbf{a} = {a_i \choose a_j}
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Notation

  • Scalars are written as normal text, for example aia_i
  • Vectors are often written in bold, underline or with an arrow on top, for example a,  a  or  a\mathbf{a}\text{,}\; \underline{a}\;\text{or}\; \overrightarrow{a}.
  • When doing your working out, be sure to pick a notation and stick with it.
    • We suggest using the underline notation, as it is the quickest and easiest to write.
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Magnitude

  • The magnitude of a vector written in component form is found by using Pythagoras' theorem:
    • a=ai2+aj2|\mathbf{a}| = \sqrt{{a_i}^2 + {a_j}^2}
  • Where a|\mathbf{a}| is the magnitude of the vector and ai,  aja_i\text{,}\;a_{j} are the x and yx\text{ and }y-components of the vector.
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Direction

  • The direction of the vector can be worked out by using trigonometry:
    • tanθ=ajai\text{tan}\,\theta = \frac{\large a_j}{\large a_i}
  • Where θ\theta is the angle the vector a\mathbf{a} makes with the horizontal.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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