Study guide and
9 practice problems
on:
Direction of a vector
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Study Guide
The direction of a vector $\mathbf{x}$ is defined by $\text{dir } \mathbf{x} = \frac{\mathbf{x}}{| \mathbf{x} | }.$
(4 problems)
To construct $\bfx$ from its length and direction, write: $\bfx = \left| \bfx \right| \text{ dir } \bfx.$
(4 problems)
Related topics
Vectors
(55 problems)
Multivariable calculus
(147 problems)
Practice problems
A river flows with speed $10$ m/s in the northeast direction. A particular boat can propel itself at speed $20$ m/s relative to the water. In which direction should the boat point in order to travel due west.?
Solution
Find the vector of length 2 in the direction of $\langle 1,-1 \rangle$.
Solution
Let $c$ be a nonzero scalar. Does $c \ \mathbf{x}$ have the same direction as $\mathbf{x}$?
Solution
If $\text{dir } \mathbf{x} = \text{dir } \mathbf{y}$ show that $|\mathbf{x} + \mathbf{y}| = |\mathbf{x}| + | \mathbf{y} |$.
Solution
Find a unit vector in the direction of $\langle 1, 1 \rangle$.
Solution
Compute the vector projection of $\bfi$ onto $\bfi + \bfj$.
Solution
Find a unit vector perpendicular to $\langle 1, 1, 1\rangle$ and $\langle 1, 0, 1 \rangle.$
Solution
Let $f(x,y) = x^2 + y^3 $ and $P=(1,1)$.
Find a direction at $P$ along which $f$ is not changing.
Solution
Write down the vector field $\mathbf{F}(x,y)$ whose value at $(x,y)$ is of unit length and points toward $(1,0)$.
Solution