Answer:
Unit Vector in the Direction of a Given Vector
Recall that a unit vector is a vector of length 1. Given a non-zero vector , we can find a unit vector in the same direction by multiplying by an appropriate scalar. For example, if and , then a unit vector in the same direction is given by .
In general, dividing a non-zero vector by its own magnitude produces a unit vector in the same direction. We summarize this observation in a theorem.
Let be a non-zero vector in . Vector given by \begin{equation*} \vec{u}=\frac{1}{\norm{\vec{v}}}\vec{v}=\begin{bmatrix}v_1/\norm{\vec{v}}\\v_2/\norm{\vec{v}}\\\vdots \\v_n/\norm{\vec{v}}\end{bmatrix} \end{equation*}
is a unit vector in the direction of .
- Proof
- Because is a positive scalar multiple of , points in the direction of . We
now show that . \begin{eqnarray*} \norm{\vec{u}}&=&\sqrt{ \Big (\frac{v_1}{\norm{\vec{v}}}\Big )^2+\Big (\frac{v_2}{\norm{\vec{v}}}\Big )^2+\ldots +\Big (\frac{v_n}{\norm{\vec{v}}}\Big )^2}\\ &=&\frac{1}{\norm{\vec{v}}}\sqrt{v_1^2+v_2^2+\ldots +v_n^2}\\ &=&\frac{\norm{\vec{v}}}{\norm{\vec{v}}}=1 \end{eqnarray*}
Practice Problems
Problems prob:00361-prob:00363
Find a unit vector in the direction of the given vector .
Let . Apply the concepts from this section to find a vector that points in the same
direction as and whose length is 5.
2024-09-11 17:59:28 Answer: