10.1.3

Vectors - Algebraic Operations

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Adding and Subtracting Vectors

Writing vectors in component form makes adding and subtracting vectors very simple.

Illustrative background for AdditionIllustrative background for Addition ?? "content

Addition

  • If we have two vectors a\mathbf{a} and b\mathbf{b}, we can write their sum as:
    • a+b=(aii+ajj)+(bii+bjj)=(ai+bi)i+(aj+bj)j\mathbf{a} + \mathbf{b} = (a_i\mathbf{i} + a_j\mathbf{j}) + (b_i\mathbf{i} + b_j\mathbf{j}) = (a_i + b_i)\mathbf{i} + (a_j + b_j)\mathbf{j}
  • Where ai,bia_i,b_i are the i\mathbf{i} components of a,b\mathbf{a},\mathbf{b}, and likewise for j\mathbf{j}.
  • In column form this is:
    • a+b=(aiaj)+(bibj)=(ai+biaj+bj)\mathbf{a} + \mathbf{b} = {a_i\choose a_j}+{b_i \choose b_j} = {a_i + \,b_i \choose a_j+\,b_j}
Illustrative background for SubtractionIllustrative background for Subtraction ?? "content

Subtraction

  • If we have two vectors a\mathbf{a} and b\mathbf{b}, we can write their difference as:
    • ab=(aii+ajj)(bii+bjj)=(aibi)i+(ajbj)j\mathbf{a} - \mathbf{b} = (a_i\mathbf{i} + a_j\mathbf{j}) - (b_i\mathbf{i} + b_j\mathbf{j}) = (a_i - b_i)\mathbf{i} + (a_j - b_j)\mathbf{j}
  • In column form this is:
    • ab=(aiaj)(bibj)=(aibiajbj)\mathbf{a} - \mathbf{b} = {a_i\choose a_j}-{b_i \choose b_j} = {a_i -\, b_i \choose a_j-\,b_j}

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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