VECTORS.

Contents:

 

Notation & Representation: A vector is a quantity made up of magnitude (size) and direction. Examples of vectors are: displacement, acceleration, force and momentum. The wind is an example of a vector, it has distance and direction. A scalar is magnitude only, examples are distance, time, speed, area, etc. To be able to operate with vectors a knowledge of Pythagorus' Theroem, and Trigonometry is necessary.

A vector may be represented by an arrowed line on a diagram: . The length of the line represents the magnitude and the arrow represents the direction. The vector may be named:, the letters denoting the end points, the arrow denoting direction from A to B. The |AB| may be used to represent the magnitude or scalar. The vector may also be named in the diagram using a small bold typed letter:. A small letter may also represent a vector when underlined: a, and the magnitude represented by: |a| or a.

A Unit Vector is a vector with a magnitude of 1. A null vector is a vector with a magnitude of zero.

Displacement Vector: Is the movement from one position to another. A displacement vector can be

represented by a column vector. The distance of x over the distance of y. x and y are the components of the vector.

The magnitude |a| = x+ y.

A position vector describes the movement from an origin to a point on a co-ordinate grid system. The position of P is given in relation to the origin O.

The inverse of a vector a is -a, equal in magnitude but opposite direction.

The inverse of a vector is -or .

Equal or equivalent vectors are equal in magnitude and direction making them parallel.

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Multiplication by a Scalar: Multiplying a vector a by a scalar increases the magnitude of the vector by that scalar number.

1. vector a. 
2. vector a times 2.
3. vector a times - 3.

 

When multiplying a column vector by a scalar, multiply each part by the scalar:

ie. a = then -3a = .

Addition & Subtraction: Vectors can be combined by addition and subtraction, the new vector becomes the resultant. The resultant is usually marked with a double arrow.

Addition: a + b = c.

      += .

Subtraction: a - b = c.

- = .

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Column Vectors: Addition and subtraction done as above, beware of the signs.

Example: p = and q = .

Find (i), p + q. Answer: p + q. = + = .

Find (ii), q - p. Answer: q - p. = - = .

Find (iii), 3p - 2q. Answer: 3p - 2q. = - = .

Find (iv), |p|. Answer: |p|. = 5+ 2. = 54.

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Geometric Problems: Vectors may be used to show geometric results, usually done by finding an expression for an unknown vector in terms of other given vectors. This is done by finding a pathway through the diagram from the start point to the finish point. Always label the unknown vector with a double arrow. The diagram shows M and N as halfway points.

= -c -b -a.

= -a + b + c. or -d.

= a - d.

 

Midpoint Theorem: In the triangle ABC, X and Y are the midpoints of AB and AC. Using vectors to prove that XY is half the length of BC and parrallel to BC.

Let = a and = b.

Therefore = + . = a + b.

= 2XA, = 2a, and = 2AY, = 2b.

Therefore = + = 2a + 2b.

Finally: = 2. BC is twice the length of XY and are parallel as they are multiples of the same vector: a + b : 2a + 2b.

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Applied Example:

The width of the river is 50metres, a swimmer swims across at the speed of 2metres per second (m/s), the current is 3m/s. Find the overall speed, the direction from the start point, and the distance the swimmer swam to reach the far bank.

Answer: (i) Overall speed. The speed of the swimmer and current are needed. Magnitude |resultant| = 2+ 3 = 3 6m/s.

Answer (ii) Direction. Using Trigonometry; Tan = 563.

Answer (iii) Distance. Using Trigonometry; cos 563 = 50 distance. Therefore distance swam = 50 cos 563. = 901 metres.

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