Questions and Answers
What is a unit vector?
V / ||V||
What is the vector formula relating to magnitude and theta in terms of i and j?
V = (||V||)(cos θ)i + (||V||)(sin θ)j
What is the formula for the magnitude of a vector?
||V|| = √[(x2-x1)^2 + (y2-y1)^2]
What is the formula for projwv?
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What is v1 in terms of projwv?
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What is v2 in terms of v and v1?
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How is work calculated when the force and motion are in the same direction?
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What is the formula for work when the force and motion are not in the same direction?
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What is the formula for cos θ in terms of vectors v and w?
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How is a vector expressed in terms of two coordinates?
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What is the formula for work when the angle is 'to the horizontal'?
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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.
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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.
