Your phone's map says the stadium is 5 miles away and you need to head northeast. Two facts, how far and which way, are all a vector needs. This page turns those two facts into numbers: the magnitude (how long) and the direction (along which line, pointing which way). Some problems call the magnitude the modulus and talk about the sense of a vector. That is just the "which way along the line" part. At the end we will answer a question the map can't: how can you tell, without drawing, whether two arrows point along the same line?
Magnitude and distance
Drag the point below. Watch how the horizontal and vertical legs change while the vector stays the hypotenuse.
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Calculate the modulus of the vector $\vec{v} = (3, -4)$.
Calculate the modulus of the vector $\vec{v} = (3, -4)$.
Prove analytically whether the vectors $\vec{u} = (2, -5)$ and $\vec{v} = (-4, 10)$ have the same direction. If so, indicate whether they have the same sense or opposite…
Prove analytically whether the vectors $\vec{u} = (2, -5)$ and $\vec{v} = (-4, 10)$ have the same direction. If so, indicate whether they have the same sense or opposite senses.
Find the coordinates of a vector $\vec{v}$ knowing that its modulus is $5\sqrt{2}$ and that it has the same direction and sense as the vector $\vec{u} = (-1, 1)$.
Find the coordinates of a vector $\vec{v}$ knowing that its modulus is $5\sqrt{2}$ and that it has the same direction and sense as the vector $\vec{u} = (-1, 1)$.
Two cables pull from the top of a pole. The first cable exerts a force represented by the vector $\vec{T}1 = (120, 160)$ N. The second cable exerts a force $\vec{T}2$ of…
Two cables pull from the top of a pole. The first cable exerts a force represented by the vector $\vec{T}_1 = (120, 160)$ N. The second cable exerts a force $\vec{T}_2$ of modulus $150$ N, which has the same direction but opposite sense to $\vec{T}_1$. Find the exact components of the vector $\vec{T}_2$.
A robot, a boat, or a laser beam starts at a point $A$, heads in the direction of $\vec d$, and travels a certain distance. Where does it end up?
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A vessel is located at point $A(2, -1)$ and moves in a straight line a distance of $13$ nautical miles. If the direction vector of its trajectory is proportional to…
A vessel is located at point $A(2, -1)$ and moves in a straight line a distance of $13$ nautical miles. If the direction vector of its trajectory is proportional to $\vec{d} = (5, 12)$ and it moves in the same sense as $\vec{d}$, determine the exact coordinates of the arrival point.
An explorer robot is initially located at point $A(-3, 2)$. It is programmed to advance in a straight line in the direction and sense of the vector $\vec{u} = (2, -1)$…
An explorer robot is initially located at point $A(-3, 2)$. It is programmed to advance in a straight line in the direction and sense of the vector $\vec{u} = (2, -1)$ until it reaches a point $B$, such that the distance $AB$ is equal to $4\sqrt{5}$. Next, it turns and advances in the direction and sense of the vector $\vec{w} = (1, 2)$ a distance of $3\sqrt{5}$ until it stops at point $C$. Calculate the exact coordinates of the points $B$ and $C$.
A light signal starts from point $A(0, 4)$ with a direction vector $\vec{d} = (2, -1)$. Upon reaching the x-axis at a point $P$, it reflects, changing its direction and…
A light signal starts from point $A(0, 4)$ with a direction vector $\vec{d} = (2, -1)$. Upon reaching the x-axis at a point $P$, it reflects, changing its direction and sense according to the new direction vector $\vec{d}' = (2, 1)$. If the signal continues its path until it hits the vertical line $x = 12$ at a point $Q$, calculate the total distance travelled by the light from $A$ to $Q$.
Forces, velocities, and displacements are vectors, and combining them means adding them component by component. The sum is called the resultant.
Two perpendicular forces, one horizontal and one vertical, form the legs of a right triangle, and the resultant is the hypotenuse. Change the legs below and compare the hypotenuse with the sum of the two legs.
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A chess player moves their pawn from the square represented by point $(1, 2)$ to square $(1, 4)$, and then moves a knight from $(1, 4)$ to $(3, 5)$. Calculate the…
A chess player moves their pawn from the square represented by point $(1, 2)$ to square $(1, 4)$, and then moves a knight from $(1, 4)$ to $(3, 5)$. Calculate the modulus of the total displacement vector of the piece from its initial position to its final position.
Two coplanar forces represented by the vectors $\vec{F}1 = (30, -40)$ N and $\vec{F}2 = (-15, 20)$ N act on a body. Calculate the modulus of the resultant force…
Two coplanar forces represented by the vectors $\vec{F}_1 = (30, -40)$ N and $\vec{F}_2 = (-15, 20)$ N act on a body. Calculate the modulus of the resultant force $\vec{R} = \vec{F}_1 + \vec{F}_2$ and determine a unit vector in the direction of this resultant.
Three coplanar forces keep an object in static equilibrium at the origin of coordinates (i.e., the resultant force is the zero vector). The components of two of them are…
Three coplanar forces keep an object in static equilibrium at the origin of coordinates (i.e., the resultant force is the zero vector). The components of two of them are known: $\vec{F}_1 = (40, 30)$ N and $\vec{F}_2 = (-20, 50)$ N. Calculate the components of the third force $\vec{F}_3$ and its exact modulus.
Often a vector or point contains an unknown, and the problem gives you a distance or magnitude. The plan is always the same:
Write the vector (or the vector between the two points) with the unknown inside.
Set $|\vec v|^2 = (\text{given magnitude})^2$. Squaring removes the square root.
Solve the equation. Usually it is quadratic, so there are often two solutions.
Check each solution in the original problem.
Some problems say rectilinear distance. That just means the ordinary straight-line distance $d(A,B)$ from this page.
If a condition links the components, for example "the two components add up to 4", substitute: $y = 4 - x$, then put that into $x^2 + y^2 = (\text{magnitude})^2$.
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A drone moves in a straight line according to the vector $\vec{d} = (k, 6)$. If the total distance traveled is $10$ feet, calculate the possible values of the parameter…
A drone moves in a straight line according to the vector $\vec{d} = (k, 6)$. If the total distance traveled is $10$ feet, calculate the possible values of the parameter $k$.
An airplane flies with an airspeed given by the vector $\vec{v}a = (250, 150)$ mph. In that area, the wind blows with a velocity represented by $\vec{v}v = (k, -50)$…
An airplane flies with an airspeed given by the vector $\vec{v}_a = (250, 150)$ mph. In that area, the wind blows with a velocity represented by $\vec{v}_v = (k, -50)$ mph. If we know that the actual velocity of the airplane with respect to the ground, calculated as $\vec{v}_r = \vec{v}_a + \vec{v}_v$, has an exact modulus of $300$ mph, determine the possible values of the parameter $k$.
Consider a parallelogram $ABCD$ with three known vertices: $A(0, 0)$, $B(6, 2)$ and $D(2, 4)$. Work out the coordinates of vertex $C$, find the modulus of its two…
Consider a parallelogram $ABCD$ with three known vertices: $A(0, 0)$, $B(6, 2)$ and $D(2, 4)$. Work out the coordinates of vertex $C$, find the modulus of its two diagonals and justify analytically whether or not it is a rhombus (i.e., whether all its sides are equal).
Back to the start: to know if two arrows lie along the same line, compare their slopes or test $u_1v_2 - u_2v_1 = 0$. To know how far you are from the stadium and which way to go, give the magnitude and a unit vector.
Let's practice! 8 medium exercises in this section