How to tell if 2 vectors are parallel?
Understand the Problem
The question is asking for the criteria or methods to determine if two vectors are parallel. This involves understanding the mathematical properties of vectors and possibly using the dot product or the scalar multiple method.
Answer
Vectors are parallel if they are scalar multiples of each other, if their cross product is zero, or if the angle between them is \(0^\circ\) or \(180^\circ\).
Answer for screen readers
Vectors ( \mathbf{a} ) and ( \mathbf{b} ) are parallel if:
- One vector is a scalar multiple of the other.
- The cross product ( \mathbf{a} \times \mathbf{b} = \mathbf{0} ) in 3D.
- The angle between them is ( 0^\circ ) or ( 180^\circ ).
Steps to Solve
- Identify Vector Representation
To determine if two vectors are parallel, first represent them as vectors. Let vector ( \mathbf{a} ) be represented as ( \mathbf{a} = \langle a_1, a_2 \rangle ) and vector ( \mathbf{b} ) as ( \mathbf{b} = \langle b_1, b_2 \rangle ).
- Scalar Multiplication Method
Vectors ( \mathbf{a} ) and ( \mathbf{b} ) are parallel if one is a scalar multiple of the other. This means there exists a scalar ( k ) such that:
$$ \mathbf{b} = k \cdot \mathbf{a} $$
This can be written as:
$$ b_1 = k \cdot a_1 $$ $$ b_2 = k \cdot a_2 $$
- Cross Product Method (in 3D)
For 3D vectors, another method is to use the cross product. Vectors ( \mathbf{a} = \langle a_1, a_2, a_3 \rangle ) and ( \mathbf{b} = \langle b_1, b_2, b_3 \rangle ) are parallel if:
$$ \mathbf{a} \times \mathbf{b} = \mathbf{0} $$
This means the cross product results in the zero vector.
- Angle Between Vectors
Two vectors are parallel if the angle ( \theta ) between them is 0 or 180 degrees. This can be determined using the dot product:
$$ \mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | | \mathbf{b} | \cos(\theta) $$
If ( \cos(\theta) = \pm 1 ), then ( \theta = 0^\circ ) or ( \theta = 180^\circ ), indicating parallel vectors.
Vectors ( \mathbf{a} ) and ( \mathbf{b} ) are parallel if:
- One vector is a scalar multiple of the other.
- The cross product ( \mathbf{a} \times \mathbf{b} = \mathbf{0} ) in 3D.
- The angle between them is ( 0^\circ ) or ( 180^\circ ).
More Information
Parallel vectors point in the same or exactly opposite directions and can be considered as extending infinitely in those directions. Understanding parallelism aids in various applications in physics and engineering.
Tips
- Confusing conditions for orthogonality (perpendicular vectors) with parallel vectors.
- Not considering the possibility of negative scaling, which can also result in parallel vectors when one vector is the negative of another.
