Study guide and
10 practice problems
on:
Vector addition
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Study Guide
$\langle x_1, x_2\rangle + \langle y_1, y_2\rangle = \langle x_1+y_1, x_2+y_2\rangle$.
(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)
Related topics
Vectors
(55 problems)
Multivariable calculus
(147 problems)
Practice problems
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
The vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are shown below. Using only vector addition and subtraction, express $\mathbf{b}$ in terms of $\mathbf{a}$, $\mathbf{c}$, and $\mathbf{d}$.
Solution
Estimate what positive multiples of the following vectors add up to zero.
Solution
If $\text{dir } \mathbf{x} = \text{dir } \mathbf{y}$ show that $|\mathbf{x} + \mathbf{y}| = |\mathbf{x}| + | \mathbf{y} |$.
Solution
Let $\mathbf{z}$ be the point one third of the way from $\mathbf{x}$ to $\mathbf{y}$. Using vector arithmetic, express $\mathbf{z}$ in terms of $\mathbf{x}$ and $\mathbf{y}$.
Solution
Use vectors and dot products to prove: if the diagonals of a rectangle are perpendicular, then the rectangle is a square.
Solution
Prove the parallelogram law: The sum of the squares of the lengths of both diagonals of a parallelogram equals the sum of the squares of the lengths of all four sides.
Solution
Prove for 2d vectors: $\bfa \cdot (\bfb + \bfc) = \bfa\cdot \bfb + \bfa \cdot \bfc.$
Solution