Study guide and
20 practice problems
on:
Vector subtraction
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Study Guide
$\langle y_1, y_2\rangle - \langle x_1, x_2\rangle = \langle y_1-x_1, y_2-x_2\rangle$.
(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)
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
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
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
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
Suppose that two opposite sides of a quadrilateral are parallel and have equal length. Show that the quadrilateral is a parallelogram.
Solution
Consider an arbitrary quadrilateral. The two blue line segments connect the midpoints of adjacent sides. Using only vector addition and multiplication by constants, show that these line segments are parallel and have the same length.
Solution
Show that the line connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.
Solution
Use vector addition, subtraction, and scalar multiplication to show that the midpoint between the two points $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.
Solution
Let $\mathbf{O}, \mathbf{N}, \mathbf{B}$ be three points as shown.
(a) What is the sign of the dot product $(\mathbf{B} - \mathbf{O}) \cdot (\mathbf{N} - \mathbf{O})$?
(b) What is the sign of the dot product $(\mathbf{B} - \mathbf{N}) \cdot (\mathbf{N} - \mathbf{O})$?
Solution
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