how do you determine if two vectors are
a) Collinear
b) orthogonal?
a) if one is a multiple of the other
b) if their dot product is zero
To determine if two vectors are collinear, you need to check if one vector is a scalar multiple of the other. In other words, if you can scale one vector by a constant factor and obtain the other vector, then they are collinear.
Here's the step-by-step process to determine if two vectors are collinear:
1. Take two vectors, let's say vector A and vector B.
A = (a1, a2, a3)
B = (b1, b2, b3)
2. Compute the ratio of the corresponding elements of the vectors:
If A = kB, where k is a scalar constant, then:
a1/b1 = a2/b2 = a3/b3 = k
3. If the ratio is the same for all the corresponding elements, then vectors A and B are collinear. Otherwise, they are not collinear.
To determine if two vectors are orthogonal, you need to check if their dot product is zero. The dot product of two vectors is calculated by taking the sum of the products of their corresponding elements.
Here's the step-by-step process to determine if two vectors are orthogonal:
1. Take two vectors, let's say vector A and vector B.
A = (a1, a2, a3)
B = (b1, b2, b3)
2. Compute the dot product of A and B:
Dot product(A, B) = a1 * b1 + a2 * b2 + a3 * b3
3. If the dot product is zero, then vectors A and B are orthogonal. Otherwise, they are not orthogonal.
So, to summarize:
- Two vectors are collinear if one vector is a scalar multiple of the other.
- Two vectors are orthogonal if their dot product is zero.