PreCalc Vectors Formulas Flashcards

Study Notes

Unit Vectors

A unit vector is defined as the vector ( V ) divided by its magnitude ( ||V|| ).

Vector Representation in Terms of Angle

The vector can be expressed as ( V = (||V||)(\cos \theta)i + (||V||)(\sin \theta)j ), where ( \theta ) is the angle with respect to the x-axis.

Magnitude Calculation

The magnitude of a vector ( V ) connecting points ( (x1, y1) ) and ( (x2, y2) ) is calculated using the formula ( ||V|| = \sqrt{(x2-x1)^2 + (y2-y1)^2} ).

Projection of Vector

The formula for the projection of vector ( v ) onto vector ( w ) is given by ( \text{proj}_{w}v = \frac{(v \cdot w)}{||w||^2} w ).

Components of Projection

The vector ( v1 ) is determined by the projection of ( v ) onto ( w ), hence ( v1 = \text{proj}_{w}v ).
The component of vector ( v ) orthogonal to ( w ) is defined as ( v2 = v - v1 ).

Work Calculations

Work when force and motion are in the same direction is calculated using the formula: [ \text{Work} = (\text{magnitude of force})(\text{distance of motion}) ]
When the force and the distance vector are not in the same direction, work is calculated using: [ W = ||F|| \cdot ||\text{distance vector}|| \cdot \cos \theta ]

Cosine of Angle

The cosine of the angle ( \theta ) between two vectors ( v ) and ( w ) is expressed as: [ \cos \theta = \frac{(v \cdot w)}{||v|| , ||w||} ]

Vector in Two-Dimensional Plane

A vector ( v ) in terms of two coordinates is given by: [ v = (x2-x1)i + (y2-y1)j ]

Work Calculation at an Angle

To calculate work when force is at an angle to the horizontal:
- Determine the vector representation of force using ( \cos ) and ( \sin ).
- Find the vector ( AB ) using coordinates.
- Finally, use the dot product of the two vectors to find the work done.

Description

Test your understanding of vector formulas with this set of flashcards. Each card covers key concepts such as unit vectors, vector magnitude, and projections. Perfect for students looking to reinforce their knowledge in precalculus.