Study guide and
32 practice problems
on:
Length of a vector
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Study Guide
The length of the vector $\mathbf{x} = \langle x_1, x_2, x_3 \rangle$ is $
|\mathbf{x}| = \sqrt{x_1^2 + x_2^2 + x_3^2}$.
(16 problems)
For any scalar $c$ and vector $\mathbf{x}$, $|c \mathbf{x} | = |c|\ |\mathbf{x} |$.
(4 problems)
A unit vector is a vector with length 1.
(5 problems)
The find the unit vector in the same direction as $\bfx$, divide $\bfx$ by its length.
(3 problems)
Related topics
Vectors
(55 problems)
Multivariable calculus
(147 problems)
Practice problems
Find the length of the 2d vector $2 \ \bfi + 3 \ \bfj$ and the 3d vector $\langle2, 3, 4 \rangle$.
Solution
Sketch all the unit vectors in 3d that have an angle of $\pi/6$ with respect to the vector $\bfi$.
Solution
Sketch all the unit vectors in 2d that have an angle of $\pi/4$ with respect to the vector $\bfi$.
Solution
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
Consider a pyramid with square base formed by the points $(1,1,0), (1,-1, 0), (-1, 1, 0), (-1, -1, 0),$ and $(0,0,1)$. What is the length of each edge connecting the base to the apex?
Solution
Find the length of the vector from $(2,4,5)$ to $(3, -1, -2)$.
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
For what value(s) of $c$ is $c (\textbf{i} + \textbf{j} + \textbf{k})$ a unit vector?
Solution
If $\text{dir } \mathbf{x} = \text{dir } \mathbf{y}$ show that $|\mathbf{x} + \mathbf{y}| = |\mathbf{x}| + | \mathbf{y} |$.
Solution
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