Emma's delivery drone takes off from a warehouse and lands at a customer's house. Knowing where it started and where it ended isn't enough for the pilot. She also needs to know how far it moved east, how far it moved north, and how far it flew in a straight line. One new object stores all of that: the vector.
Notation: in this page, points are written with parentheses, like $A(2, 5)$, and vectors with angle brackets, like $\langle 3, 4 \rangle$. Some problems write vectors with parentheses, as $(3, 4)$. They mean the same thing.
Vectors and Components
A character in a video game walks from $A(-1, 1)$ to $B(3, 4)$ on the map. The change in $x$ is $3 - (-1) = 4$ and the change in $y$ is $4 - 1 = 3$. So $\vec{AB} = \langle 4, 3 \rangle$.
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Given the points $A(2, -1)$ and $B(5, 3)$, calculate the components of the vector $\vec{AB}$ and its magnitude.
Given the points $A(2, -1)$ and $B(5, 3)$, calculate the components of the vector $\vec{AB}$ and its magnitude.
The magnitude of a vector is its length: the straight-line distance from start to end. The components are the legs of a right triangle, so the magnitude is the hypotenuse.
Drag the point below and watch the right triangle. The distance from the origin is the magnitude of the vector $\langle x, y \rangle$.
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Find the possible values of $x$ so that the vector $\vec{v} = (x, -8)$ has a magnitude exactly equal to $10$.
Find the possible values of $x$ so that the vector $\vec{v} = (x, -8)$ has a magnitude exactly equal to $10$.
Geometrically, to add vectors you place them tip to tail. The sum goes from the start of the first to the end of the second.
Velocities are vectors too. A kayaker paddles $\langle 4, 0 \rangle$ mph east while the current pushes $\langle 0, -3 \rangle$ mph south. The actual velocity is the sum, $\langle 4, -3 \rangle$ mph. The speed is its magnitude, $\sqrt{16 + 9} = 5$ mph.
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Consider the vectors $\vec{u} = (3, -2)$ and $\vec{v} = (-1, 4)$. Calculate the components of the resultant vector of the operation $\vec{w} = 2\vec{u} - 3\vec{v}$.
Consider the vectors $\vec{u} = (3, -2)$ and $\vec{v} = (-1, 4)$. Calculate the components of the resultant vector of the operation $\vec{w} = 2\vec{u} - 3\vec{v}$.
Observe the vectors $\vec{u}$ and $\vec{v}$ represented on the grid. Write down their components and calculate analytically the coordinates of the sum vector…
Observe the vectors $\vec{u}$ and $\vec{v}$ represented on the grid. Write down their components and calculate analytically the coordinates of the sum vector $\vec{s} = \vec{u} + \vec{v}$.
A delivery drone flies with its own velocity represented by the vector $\vec{v}d = (12, 5)$ mph. At that moment, the wind blows with a velocity of $\vec{v}v = (-3, 2)$…
A delivery drone flies with its own velocity represented by the vector $\vec{v}_d = (12, 5)$ mph. At that moment, the wind blows with a velocity of $\vec{v}_v = (-3, 2)$ mph. Calculate the actual velocity vector of the drone and its speed (the magnitude of the resultant vector).
The total length of a segment $AB$ is $\lVert \vec{AB} \rVert$. For problems that ask for the midpoint and the length, do both calculations separately.
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Determine the coordinates of the midpoint $M$ of the line segment whose endpoints are the points $P(-4, 7)$ and $Q(2, -3)$.
Determine the coordinates of the midpoint $M$ of the line segment whose endpoints are the points $P(-4, 7)$ and $Q(2, -3)$.
The point $M(1, 2)$ is the midpoint of the line segment $AB$. If we know the coordinates of the endpoint $A(-3, 5)$, find the coordinates of the endpoint $B$.
The point $M(1, 2)$ is the midpoint of the line segment $AB$. If we know the coordinates of the endpoint $A(-3, 5)$, find the coordinates of the endpoint $B$.
A straight pipe connects a pumping station $A(-15, 20)$ with a tank $B(45, -12)$ (coordinates in feet). A control valve is to be installed exactly at the midpoint of the…
A straight pipe connects a pumping station $A(-15, 20)$ with a tank $B(45, -12)$ (coordinates in feet). A control valve is to be installed exactly at the midpoint of the route. Calculate the coordinates where the valve will be located and the total length of the pipe.
A point that is a fraction $t$ of the way from $A$ to $B$ is found by starting at $A$ and adding that fraction of the vector:
$$P = A + t\,\vec{AB}$$
For the midpoint, $t = \tfrac12$. To cut a segment into 3 equal pieces, use $t = \tfrac13$ and $t = \tfrac23$.
The centroid of a triangle is the point where the three medians meet. It lies $\tfrac23$ of the way from each vertex to the midpoint of the opposite side. So $G = A + \tfrac23\vec{AM}$, where $M$ is the midpoint of $\overline{BC}$.
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A steel beam whose ends are at the coordinates $A(-4, 2)$ and $B(8, 14)$ must be cut into three identical parts to manufacture supports. Calculate the coordinates of the…
A steel beam whose ends are at the coordinates $A(-4, 2)$ and $B(8, 14)$ must be cut into three identical parts to manufacture supports. Calculate the coordinates of the two cutting points $P$ and $Q$ that divide the line segment $AB$ into three parts of equal length.
The centroid $G$ (center of gravity) of a triangle is always located at a distance of $\frac{2}{3}$ from any vertex to the midpoint of the opposite side. If the vertices…
The centroid $G$ (center of gravity) of a triangle is always located at a distance of $\frac{2}{3}$ from any vertex to the midpoint of the opposite side. If the vertices of a triangular metal plate are $A(1, 1)$, $B(7, 3)$ and $C(4, 8)$, first find the midpoint $M$ of the side $BC$. Then, using the vector $\vec{AM}$, calculate the exact coordinates of the centroid $G$.
Two vectors are parallel if one is a scalar multiple of the other: $\vec u = k\vec v$. If $k > 0$ they point the same way. If $k < 0$ they point opposite ways.
Collinear points. Three points $A$, $B$, $C$ lie on one line exactly when $\vec{AB}$ and $\vec{AC}$ are parallel. Both vectors start at $A$, so being parallel means they lie on the same line.
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State whether the vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-2, 3)$ are parallel (they have the same direction). Justify your answer mathematically.
State whether the vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-2, 3)$ are parallel (they have the same direction). Justify your answer mathematically.
In the design of a bridge, three pillars must be perfectly aligned. On the architect's plan, the bases of the pillars are located at the coordinates $P(-2, 1)$,…
In the design of a bridge, three pillars must be perfectly aligned. On the architect's plan, the bases of the pillars are located at the coordinates $P(-2, 1)$, $Q(2, 3)$ and $R(8, k)$. Determine the value of $k$ so that the alignment is correct.
On a billiard table, a ball is struck and follows the trajectory of the vector $\vec{u} = (4, 3)$. After bouncing off the cushion, its new trajectory is given by the…
On a billiard table, a ball is struck and follows the trajectory of the vector $\vec{u} = (4, 3)$. After bouncing off the cushion, its new trajectory is given by the vector $\vec{v} = (-3, k)$. Calculate the value of $k$ so that the rebound trajectory forms an angle of $90^\circ$ with the arrival trajectory.
Find out if the triangle formed by the telecommunications antennas located at the coordinates $A(1, 2)$, $B(4, 6)$ and $C(9, -1)$ is a right triangle. Use the scalar…
Find out if the triangle formed by the telecommunications antennas located at the coordinates $A(1, 2)$, $B(4, 6)$ and $C(9, -1)$ is a right triangle. Use the scalar product of the vectors that form its sides to justify your answer.
Parallelograms. In parallelogram $ABCD$, opposite sides are equal vectors: $\vec{AB} = \vec{DC}$ and $\vec{AD} = \vec{BC}$. To find a missing vertex, start at $A$ and walk along $\vec{BC}$: $D = A + \vec{BC}$.
Squares. If two opposite vertices of a square are $A$ and $B$, then $\lVert \vec{AB} \rVert$ is the diagonal $d$, the center is the midpoint of $\overline{AB}$, and the area is $\dfrac{d^2}{2}$. This is because a side $s$ satisfies $d = s\sqrt2$, so $d^2 = 2s^2$. For example, with $(0, 1)$ and $(4, -3)$: $d^2 = 16 + 16 = 32$, the area is $16$, and the center is $(2, -1)$.
Mirrors. Reflecting a point across the $x$-axis changes the sign of $y$: $(x, y) \to (x, -y)$. In a mirror problem, a reflected path looks like a straight line from the image of the source.
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To install a solar panel in the shape of a parallelogram $ABCD$, the technicians have fixed three of its vertices at the coordinates $A(1, 1)$, $B(5, 2)$ and $C(6, 6)$…
To install a solar panel in the shape of a parallelogram $ABCD$, the technicians have fixed three of its vertices at the coordinates $A(1, 1)$, $B(5, 2)$ and $C(6, 6)$ (in feet). Calculate the coordinates of the fourth vertex $D$ using vector operations.
A laser beam starts from a source located at $F(-3, -2)$ and reflects off a flat mirror that coincides with the x-axis. If the exact point of impact on the mirror is…
A laser beam starts from a source located at $F(-3, -2)$ and reflects off a flat mirror that coincides with the x-axis. If the exact point of impact on the mirror is $I(1, 0)$, find the coordinates of the symmetrical point of $F$ with respect to the mirror and write down the components of the vector that defines the reflected trajectory.
Four buoys delimit a marine reserve zone at the points $A(-2, -1)$, $B(4, 1)$, $C(5, 5)$ and $D(-1, 3)$. Show first, using vectors, that the figure formed is a…
Four buoys delimit a marine reserve zone at the points $A(-2, -1)$, $B(4, 1)$, $C(5, 5)$ and $D(-1, 3)$. Show first, using vectors, that the figure formed is a parallelogram. Then, calculate the scalar product of its diagonals to check if it is specifically a rhombus.
Many harder problems give a condition and ask for an unknown number or point. The recipe is always the same: write the unknown into the coordinates, translate the condition into an equation, and solve.
Equidistant points. A point $P$ is the same distance from $A$ and $B$ when $PA = PB$. All such points lie on the perpendicular bisector of $\overline{AB}$. Squaring both sides avoids square roots.
Parameters. If the vectors contain an unknown, use the parallel test or the dot product.
A perpendicular vector of a given length. The vector $\langle -b, a \rangle$ is perpendicular to $\langle a, b \rangle$ and has the same magnitude. Multiply by a number to get the length you need. Both $k$ and $-k$ work, so there are two answers.
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Determine the coordinates of a point $P$ located on the y-axis that is at the same distance from the points $A(2, 5)$ and $B(-4, 1)$.
Determine the coordinates of a point $P$ located on the y-axis that is at the same distance from the points $A(2, 5)$ and $B(-4, 1)$.
Consider the vectors dependent on a parameter $\vec{u} = (m, 2)$ and $\vec{v} = (3, m-1)$. Find the values of the parameter $m$ in two different geometric cases: a) So…
Consider the vectors dependent on a parameter $\vec{u} = (m, 2)$ and $\vec{v} = (3, m-1)$. Find the values of the parameter $m$ in two different geometric cases:
a) So that $\vec{u}$ and $\vec{v}$ are perpendicular.
Find the components of a vector $\vec{w}$ that is perpendicular to the vector $\vec{u} = (-5, 12)$ and whose magnitude is exactly half the magnitude of $\vec{u}$. How…
Find the components of a vector $\vec{w}$ that is perpendicular to the vector $\vec{u} = (-5, 12)$ and whose magnitude is exactly half the magnitude of $\vec{u}$. How many mathematical solutions exist for $\vec{w}$? Write them all down.
Back to the drone: if it flies from $A$ to $B$, the vector $\vec{AB}$ gives the heading through its horizontal and vertical components. Its magnitude $\lVert \vec{AB} \rVert$ is the straight-line distance flown.
Summary
Let's practice! 8 medium exercises in this section