how to determine vectors u and v are parallel, orthogonal, or neither?
If you know dot and cross products
if parallel, |A cross B| = |A| |B| sin theta
if parallel theta and sin theta = 0 and cross product is 0
if perpendicular then dot product = |A| |B| cos theta = 0
If you are not there yet
then
do y = m x + b for both
if the two m s are the same, same slope, parallel
if m for second = -1/m for first, then perpendicular
To determine whether two vectors u and v are parallel, orthogonal, or neither, you can use the concepts of dot product and cross product.
1. Parallel Vectors:
Two vectors u and v are parallel if and only if their directions are the same or opposite, regardless of their magnitudes. To check for parallelism, calculate the dot product of the vectors.
- If the dot product (u · v) is equal to the product of the magnitudes of the vectors (|u| * |v|), then u and v are parallel. The dot product is equal to the cosine of the angle between the vectors multiplied by the magnitudes of the vectors.
2. Orthogonal Vectors:
Two vectors u and v are orthogonal (perpendicular) if and only if their dot product is zero. To check for orthogonality, calculate the dot product of the vectors.
- If the dot product (u · v) is zero, then u and v are orthogonal. This means that the angle between the vectors is 90 degrees, and they do not point in the same or opposite directions.
3. Neither Parallel nor Orthogonal:
If the dot product of two vectors u and v is neither equal to the product of their magnitudes nor zero, then u and v are neither parallel nor orthogonal. In this case, the vectors have some angle between them, and they do not follow either of the above conditions.
Remember that the dot product of two vectors can be calculated as follows:
(u · v) = u1 * v1 + u2 * v2 + ... + un * vn
where u1, u2, ..., un and v1, v2, ..., vn are the components of vectors u and v, respectively.
Additionally, if the vectors are given in terms of their components (u1, u2, ..., un and v1, v2, ..., vn), you can also find the dot product using vector multiplication and addition.
In summary, you can determine whether vectors u and v are parallel, orthogonal, or neither by calculating their dot product (u · v) and checking the results against the conditions explained above.