Vectors in two dimensions Revision Notes

A translation moves every point of a shape the same distance in the same direction, without rotating or resizing it. This movement can be described by a vector, often written as \( \vec{AB} \) or simply \( \mathbf{a} \), where \( A \) is the starting point and \( B \) is the ending point of the movement.

If the vector \( \vec{AB} = (a, b) \), then translating a point \( P = (x, y) \) by this vector moves \( P \) to \( P' = (x + a, y + b) \). This means the point shifts horizontally by \( a \) units and vertically by \( b \) units.

For example, if \( \vec{AB} = (3, -2) \), then translating \( P = (1, 4) \) by \( \vec{AB} \) gives \( P' = (1 + 3, 4 - 2) = (4, 2) \).

• Translation moves a shape without rotation or resizing.
• Represented by vector \( \vec{AB} = (a, b) \).
• Coordinates transform as \( (x, y) \to (x + a, y + b) \).

Vector addition combines two vectors by adding their corresponding components. If \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), then their sum is \( \mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2) \).

This operation can be visualised as moving along \( \mathbf{a} \) and then along \( \mathbf{b} \), resulting in a combined movement.

Example: Let \( \mathbf{a} = (2, 3) \) and \( \mathbf{b} = (-1, 4) \). Then, \( \mathbf{a} + \mathbf{b} = (2 + (-1), 3 + 4) = (1, 7) \).

• Add corresponding components: \( (a_1, a_2) + (b_1, b_2) = (a_1 + b_1, a_2 + b_2) \).
• Represents combined movement of two vectors.

Vector subtraction finds the difference between two vectors by subtracting their components. For vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), \( \mathbf{a} - \mathbf{b} = (a_1 - b_1, a_2 - b_2) \).

This can be interpreted as the vector you must add to \( \mathbf{b} \) to get \( \mathbf{a} \).

Example: If \( \mathbf{a} = (5, 2) \) and \( \mathbf{b} = (3, 6) \), then \( \mathbf{a} - \mathbf{b} = (5 - 3, 2 - 6) = (2, -4) \).

• Subtract corresponding components: \( (a_1, a_2) - (b_1, b_2) = (a_1 - b_1, a_2 - b_2) \).
• Represents the vector difference between two vectors.

Multiplying a vector by a scalar (a real number) changes its length but not its direction unless the scalar is negative, which reverses the direction.

If \( \mathbf{a} = (a_1, a_2) \) and \( k \) is a scalar, then \( k\mathbf{a} = (k a_1, k a_2) \).

Example: For \( \mathbf{a} = (3, -1) \) and \( k = -2 \), \( k\mathbf{a} = (-2 \times 3, -2 \times -1) = (-6, 2) \). The vector is stretched by a factor of 2 and reversed in direction.

• Multiply each component by the scalar: \( k(a_1, a_2) = (k a_1, k a_2) \).
• Length changes by \( |k| \), direction reverses if \( k < 0 \).

Translations can be combined using vector addition. For example, translating by \( \mathbf{a} \) then by \( \mathbf{b} \) is equivalent to translating by \( \mathbf{a} + \mathbf{b} \).

Subtracting vectors can represent reversing a translation or finding the vector between two points.

Scalar multiplication can scale a translation vector, making the movement longer or shorter, or reverse it if the scalar is negative.

• Combined translations add their vectors.
• Reversing a translation uses vector subtraction or multiplying by \( -1 \).
• Scaling translations changes the distance moved.