Vector basics
Vectors are lines with direction and magnitude from a point. This point could be the origin (0, 0), or the end of another vector.
Column Vector Form
In the diagram, an object moves from the origin (0, 0) to point B. It does not follow a straight line, but we are only interested in its displacement from the origin. So, we can say it arrives at B by moving 5 squares along the x-axis, and 5 squares up the y-axis. We can write this as a column vector:
${AB}↖{→} = (\table (x_f - x_i);(y_f - y_i)) $
$= (\table 5-0;5-0) = (\table 5;5)$
where $x_f$ and $y_f$ are the final x and y positions, and $x_i$ and $y_i$ are the intial x and y positions.
Parametric Equation Form
If the length of a side of one square is a 'unit', then we can say that the object has moved 5 units along the x-axis, and 5 units up the y-axis. One unit in the x-axis we call 'i', and one unit in the y-axis we call 'j'. This allows us to write the same vector with the parameters i and j:
${AB}↖{→} = 5i + 5j$
Example
Write the path of the object in the diagram in column vector and parametric equation forms.
Column vector: ${AB}↖{→} = (\table (x_f - x_i);(y_f - y_i)) $
$= (\table 4-1.5;1-5) = (\table 2.5;-4)$
Parametric equation: ${AB}↖{→} = 2.5i - 4j$
Unit Vector
A unit vector is a vector that is one unit long. A unit vector in the x-direction is 1i, and a unit vector in the y-direction is 1j. Therefore, the vector a =[a1, a2] may also be written in the form: a1i + a2j.
The angle between two vectors is:
$$cosθ = {a1.b1 + a2.b2}/{(|u|⋅|v|)} $$The dot product of two vectors is defined as:
$$ u.v = |u|⋅|v|⋅cosθ$$The magnitude of a vector can be calculated using the fact that a vector can be broken down to two components, in the x and y direction.
Vector magnitude and angle
The length of a vector is determined using the Pythagorean Theorem:
let a be a vector a =[a1, a1], then the length is |a| = √a12 + a22.
Vectors in Real Life
Vectors describe quantities which can have magnitude and direction. These may be:
Vectors indicate the magnitude and direction of quantities, such as the forces of weight and tension in rope. In equilibrium, the sum of these force vectors equals zero.
- force
- displacement
- velocity
- acceleration
- momentum
For example, a weight suspended by two ropes has forces in equilibrium. these means there is no net force acting, so the mass does not accelerate in any direction. The forces may be drawn in a free-body diagram, where the relative strengths of the forces are indicated by the lengths of the vector, as well as their direction. if the force vectors are added together, they will result in a zero vector.
Content © Andrew Bone. All rights reserved. Created : February 12, 2014
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