prove the vector projection of -v on v is -v using the vector projection formula.
To prove that the vector projection of -v on v is -v, we can use the vector projection formula along with some basic vector properties.
The vector projection of a vector u onto a vector v can be calculated using the formula:
proj_v(u) = (u · v / ||v||^2) * v
where · denotes the dot product and ||v|| represents the magnitude of vector v.
In this case, we want to find the vector projection of -v onto v, so u = -v and v = v.
Using the formula, we have:
proj_v(-v) = ((-v) · v / ||v||^2) * v
We can simplify this equation step-by-step:
1. Dot product of -v and v:
(-v) · v = -1 * (v · v) (using scalar multiplication property)
= -1 * ||v||^2 (since v · v = ||v||^2 for any vector v)
2. Substitute the dot product back in the formula:
proj_v(-v) = ((-1 * ||v||^2) / ||v||^2) * v
3. Simplify the numerator and denominator:
proj_v(-v) = -1 * (||v||^2 / ||v||^2) * v
4. The ratio of ||v||^2 to itself is always 1:
proj_v(-v) = -1 * 1 * v
proj_v(-v) = -v
Therefore, the vector projection of -v onto v is indeed -v, as shown using the vector projection formula.