8 Questions
When adding two vectors that are going in opposite directions, what mathematical operation is used?
Subtraction
What is the resultant of two perpendicular vectors?
Vector sum
In the Pythagorean theorem for vectors, what does the hypotenuse represent?
The resultant
Which arithmetic operation is used to add vectors pointing in the same direction?
Addition
What kind of multiplication of vectors yields another vector?
Cross product
When adding vectors using the parallelogram method, what property must the vectors have?
Similar directions
Which type of product of vectors yields a scalar quantity?
Dot product
In vector subtraction, what happens when two vectors are going in opposite directions?
They are subtracted
Study Notes
Vector Addition Methods
- There are two methods to add vectors: Graphical Methods and Algebraic Methods (also referred to as component method)
- Graphical Methods involve the use of scale drawings, such as the polygon method
Graphical Methods
- For vectors in one dimension, simple addition and subtraction are used, with attention to signs
- For vectors in two dimensions, the Pythagorean Theorem can be used to find the displacement
- The parallelogram method can also be used, with vectors added "tail-to-tip"
Vector Properties
- The associative property of addition states that A + (B + C) = (A + B) + C
- The commutative law of addition states that A + B = B + A
Subtraction of Vectors
- To subtract vectors, the negative of a vector is defined, which has the same magnitude but points in the opposite direction
- Then, the negative vector is added to the original vector
Multiplication of a Vector by a Scalar
- A vector V can be multiplied by a scalar c, resulting in a vector cV with the same direction but a magnitude cV
- If c is negative, the resultant vector points in the opposite direction
Adding Vectors by Components
- To add vectors by components, one must:
- Draw a diagram and add the vectors graphically
- Choose x and y axes
- Resolve each vector into x and y components
- Calculate each component using sines and cosines
- Add the components in each direction
- Use the length and direction of the vector to find the final result
Unit Vectors
- A unit vector is a vector with a magnitude of exactly 1 and drawn in the direction of a given vector
- It lacks dimension and unit, and its only purpose is to specify a direction in space
- A given vector can be expressed as a product of its magnitude and a unit vector: 𝐴⃗ = 𝐴
Test your knowledge on graphical and algebraic methods for adding and subtracting vectors. Includes concepts such as polygon method, component method, associative property of vector addition, and commutative law of addition.
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