Vector Projection Definition: For two vectors U and V, the projection of V onto U is a vector that captures how much of V points in the direction of U. It is notated as proj(U)(V). The projection can be calculated as: proj(U)(V) = (U dot V / ||U||^2) * U For each set of vectors below, compute proj(U)(V) and plot U, V, and proj(U)(V) on the same plot: (a) U = [J], V = J U = [B], V = 4 U = [D], V = []
The Correct Answer and Explanation is:
Vector Projection Overview
The projection of vector V onto vector U is the vector that represents the component of V in the direction of U. This is mathematically expressed as:proj(U)(V)=(U⋅V∥U∥2)⋅U\text{proj}(U)(V) = \left( \frac{U \cdot V}{\|U\|^2} \right) \cdot Uproj(U)(V)=(∥U∥2U⋅V)⋅U
Where:
- U⋅VU \cdot VU⋅V is the dot product of the vectors U and V.
- ∥U∥\|U\|∥U∥ is the magnitude of the vector U, and ∥U∥2\|U\|^2∥U∥2 is its squared magnitude.
- The result is a vector parallel to U representing how much of V is in the same direction as U.
Part (a) U=[J],V=[J]U = [J], V = [J]U=[J],V=[J]
- Here, both vectors U and V are the same, so the projection of V onto U is simply V. The dot product U⋅VU \cdot VU⋅V will be the square of the magnitude of U (or V) since they are the same vector.
Given U=[J]U = [J]U=[J] and V=[J]V = [J]V=[J]:
- Dot product U⋅V=J⋅J=J2U \cdot V = J \cdot J = J^2U⋅V=J⋅J=J2
- Magnitude of U: ∥U∥=J\|U\| = J∥U∥=J
- Projection: proj(U)(V)=(J2J2)⋅[J]=[J]\text{proj}(U)(V) = \left( \frac{J^2}{J^2} \right) \cdot [J] = [J]proj(U)(V)=(J2J2)⋅[J]=[J]
Thus, the projection of V onto U is [J], the same as V.
Part (b) U=[B],V=[4]U = [B], V = [4]U=[B],V=[4]
Given U = [B] and V = [4]:
- Dot product U⋅V=B⋅4=4BU \cdot V = B \cdot 4 = 4BU⋅V=B⋅4=4B
- Magnitude of U: ∥U∥=B\|U\| = B∥U∥=B
- Projection: proj(U)(V)=(4BB2)⋅[B]=(4B)⋅[B]\text{proj}(U)(V) = \left( \frac{4B}{B^2} \right) \cdot [B] = \left( \frac{4}{B} \right) \cdot [B]proj(U)(V)=(B24B)⋅[B]=(B4)⋅[B]
So, the projection of V onto U is a vector that is parallel to U, with magnitude proportional to 4B\frac{4}{B}B4.
Part (c) U=[D],V=[]U = [D], V = []U=[D],V=[]
For this case, since V is the zero vector (indicated by []), the projection of any vector V onto any vector U is the zero vector.
- The dot product U⋅V=D⋅0=0U \cdot V = D \cdot 0 = 0U⋅V=D⋅0=0
- The projection will be: proj(U)(V)=(0D2)⋅[D]=[0]\text{proj}(U)(V) = \left( \frac{0}{D^2} \right) \cdot [D] = [0]proj(U)(V)=(D20)⋅[D]=[0]
Thus, the projection is the zero vector, [0].
Visualization
You can plot the vectors U, V, and proj(U)(V) on a graph as follows:
- For part (a), U and V are identical, so their vectors are exactly the same on the plot.
- For part (b), the projection proj(U)(V) is a scalar multiple of U, and can be plotted as a vector that points in the same direction as U but with a different magnitude.
- For part (c), V is a zero vector, and the projection will also be a zero vector, meaning no visible projection on the plot.
The general rule of thumb is that the projection vector proj(U)(V) always lies along the direction of U, regardless of V‘s direction.
