Study guide and
55 practice problems
on:
Vectors
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Study Guide
Length of a vector
(32 problems)
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)
Direction of a vector
(9 problems)
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)
Angle between vectors
(7 problems)
The angle between two vectors is the angle swept by the arc that directly connects them, provided that the vectors share the same base.
(7 problems)
The angle between vectors is always between 0 and $\pi$, inclusive. It is 0 if the vectors are in the same direction and $\pi$ if the vectors are in opposite directions.
(2 problems)
Vector addition
(10 problems)
$\langle x_1, x_2\rangle + \langle y_1, y_2\rangle = \langle x_1+y_1, x_2+y_2\rangle$.
(2 problems)
The sum of two vectors is the vector obtained by lining up the tail of one vector to the head of the other:
(6 problems)
When an object has a velocity relative to a moving medium, it's net velocity is the sum of it's relative velocity and the medium's velocity.
(1 problem)
Vector subtraction
(20 problems)
$\langle y_1, y_2\rangle - \langle x_1, x_2\rangle = \langle y_1-x_1, y_2-x_2\rangle$.
(3 problems)
The vector from $\bfx$ to $\bfy$ is given by $\bfy - \bfx$.
(14 problems)
Subtracting a vector is the same as adding the negative of the vector: $$\bfy - \bfx = \bfy + (- \bfx).$$
(1 problem)
Vector scalar multiplication
(13 problems)
$c \mathbf{x}$ is the vector obtained by multiplying the length of $\bfx$ by $c$.
(6 problems)
$c \langle x_1, x_2, x_3 \rangle = \langle c x_1, c x_2, c x_3 \rangle.$
(2 problems)
For any scalar $c$ and vector $\mathbf{x}$, $|c \mathbf{x} | = |c|\ |\mathbf{x} |$.
(4 problems)
Geometric proofs with vectors
(6 problems)
Begin a geometric proof by labeling important points with as few variables as possible.
(5 problems)
The midpoint between the two vectors $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.
(3 problems)
Ways of finding a perpendicular vector
(6 problems)
Related topics
Multivariable calculus
(147 problems)
Practice problems
Let $y = \frac{1}{2} x^2+\frac{1}{2}$. For what value of $c$ is $\mathbf{i} + c \mathbf{j}$ a tangent vector to $y(x)$ at $x=1$?
Solution
Find the length of the 2d vector $2 \ \bfi + 3 \ \bfj$ and the 3d vector $\langle2, 3, 4 \rangle$.
Solution
Sketch all 3d vectors whose angle with respect to the vector $\bfi$ is
$\pi/6$
$\pi/2$
$5\pi/6$
Solution
Sketch all the unit vectors in 3d that have an angle of $\pi/6$ with respect to the vector $\bfi$.
Solution
Estimate the angle between the following pairs of vectors:
Solution
Sketch all the unit vectors in 2d that have an angle of $\pi/4$ with respect to the vector $\bfi$.
Solution
The vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are shown below. Using only vector addition, express one of the vectors in terms of the others.
Solution
The following parallelogram has one corner at the origin. The two neighboring corners are given by vectors $\mathbf{a}$ and $\mathbf{b}$. Express the fourth corner as a vector.
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
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