Vectors

A vector is a quantity with both a size and a direction. A distance of 5 km is a scalar. A move of 5 km north-east is a vector. In three dimensions, lines, planes and the angles and distances between them all come from two products: the dot product and the vector product.

What is a vector?

A vector is drawn as an arrow. The length of the arrow is the size and the head shows the direction. A temperature of 20° is a scalar. A wind of 20 km/h from the north-west is a vector.

To calculate with a vector, write it as two numbers: the step across and the step up. A column vector writes the step across on top.

A vector records a movement, not a place. The arrow from (1, 1) to (4, 3) and the arrow from (0, 0) to (3, 2) are the same vector, because both mean 3 across and 2 up.

See Vectors and Column vectors.

How do you add and subtract vectors?

To add two vectors, add the across parts and add the up parts. On a diagram, start the second arrow where the first one ends. The sum is the arrow from the first start to the last finish.

To subtract a vector, add its negative, so each part subtracts. If a and b are the position vectors of two points, then b − a is the arrow from the first point to the second: destination minus start.

Multiplying a vector by a number is scalar multiplication. Multiplying by 3 makes the arrow three times as long in the same direction. A negative number reverses it as well. Two vectors are parallel exactly when one is a scalar multiple of the other.

See Adding vectors, Subtracting vectors and Scalar multiplication.

What is the magnitude of a vector, and what is a unit vector?

The magnitude of a vector is the length of its arrow, written |a|. The two components are the shorter sides of a right triangle and the vector is the hypotenuse, so Pythagoras gives it: $|\left(3,4\right)|=\sqrt{3^2+4^2}=5$|(3, 4)| = √(3² + 4²) = 5.

A unit vector has length 1. To make one in the direction of a, divide every component by |a|. For (3, 4) that gives $\left(\frac{3}{5},\frac{4}{5}\right)$(3/5, 4/5).

i is one step along the x-axis and j is one step along the y-axis, so 3i + 4j is the vector (3, 4). The position vector of a point is the vector from the origin O to that point, and the vector from A to B is AB = OB − OA.

See Magnitude of a vector, Unit vectors, The unit vectors i and j and Position vectors.

How do you prove a geometry result with vectors?

Write every point as a position vector. If A and B have position vectors a and b, the midpoint M of AB is reached by walking to A and then half of the gap: OM = a + ½(b − a) = ½(a + b).

In triangle OAB, let P and Q be the midpoints of OA and OB. Then PQ = ½b − ½a = ½AB. Because PQ is a multiple of AB, this one line proves that PQ is parallel to AB and half its length.

Two vectors are collinear when one is a multiple of the other, with the same multiplier in every component. Three points lie on one straight line when the vector from the first to the second is a multiple of the vector from the first to the third.

See Vector proofs with midpoints and Collinear vectors.

What does the dot product actually measure?

The dot product measures how far two vectors point the same way. Multiply the matching components and add: $\left(a_1,a_2\right)\cdot \left(b_1,b_2\right)=a_1{b}_1+a_2{b}_2$(a₁, a₂) · (b₁, b₂) = a₁b₁ + a₂b₂. The answer is a number, not a vector.

The same number is $|a||b|\cos \theta$|a| |b| cos θ, where $\theta$θ is the angle between the arrows. It is largest when the vectors point the same way, zero at a right angle, and negative past 90°.

To find the angle, divide the dot product by both lengths: $\cos \theta =a\cdot b/\left(|a||b|\right)$cos θ = a · b / (|a| |b|). Perpendicular vectors have a dot product of zero, because cos 90° = 0.

See The dot product and The angle between vectors.

How do vectors work in three dimensions?

A vector in three dimensions has a third component for depth: (2, 3, 6), or 2i + 3j + 6k, where k is one step into depth. Addition, scalar multiplication and the dot product work exactly as before.

The magnitude gains one more square under the root: $|\left(2,3,6\right)|=\sqrt{4+9+36}=7$|(2, 3, 6)| = √(4 + 9 + 36) = 7.

The distance between two points is the magnitude of the vector between them. The midpoint averages each coordinate on its own, z included.

See Vectors in three dimensions and Distance and midpoint in three dimensions.

What is the vector equation of a line?

A line is fixed by one point a on it and one direction d along it. Every point of the line is r = a + t d for some number t.

The Cartesian form removes the parameter. Make t the subject of each coordinate equation and set the three expressions equal: $\frac{\left(x-a_1\right)}{d_1}=\frac{\left(y-a_2\right)}{d_2}=\frac{\left(z-a_3\right)}{d_3}$(x − a₁)/d₁ = (y − a₂)/d₂ = (z − a₃)/d₃. The numerators give the point and the denominators give the direction.

In space, two lines that are not parallel can still miss each other, like a bridge and the road under it. Such lines are skew. Set the two positions equal, solve two of the three component equations for t and s, and substitute into the third. If it holds, the lines meet. If it fails, they are skew.

See The vector equation of a line, The Cartesian form of a line and Intersecting and skew lines.

What is the cross product for?

The vector product, or cross product, a × b is a new vector at right angles to both a and b: $a\times b=\left(a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1\right)$a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Swapping the order reverses the answer: b × a = −(a × b).

Its length is $|a||b|\sin \theta$|a| |b| sin θ, the area of the parallelogram the two vectors span. Half of that is the area of the triangle with the same two sides.

See The vector product and Parallelogram area from the vector product.

How do you write the equation of a plane?

From one point a on the plane and two directions b and c inside it, every point is $r=a+\lambda b+\mu c$r = a + λb + μc. The other form uses a normal n, a vector at right angles to the plane. For any point r of the plane, r − a lies in the plane, so (r − a) · n = 0, which is r · n = a · n. Call that constant d: r · n = d.

Writing r as (x, y, z) opens the dot product out into ax + by + cz = d, and the coefficients are the components of the normal. The plane 2x + 3y + z = 7 has normal (2, 3, 1).

The angle between two planes is the angle between their normals.

The angle between a line and a plane is 90° minus the angle between the line and the normal. Complementary angles swap cosine for sine, so $\sin \theta =|d\cdot n|÷\left(|d||n|\right)$sin θ = |d · n| ÷ (|d| |n|). The common mistake is to stop at $\cos \theta$cos θ, which is the angle to the normal.

See The vector equation of a plane, The Cartesian equation of a plane, The angle between two planes and The angle between a line and a plane.

How do you find where things meet, and how far apart they are?

To find where a line meets a plane, substitute the line's x, y and z, each in terms of t, into the plane equation. Solve for t, then put t back into the line. If the t terms cancel, the line is parallel to the plane.

Two planes that are not parallel meet along a line. The line lies in both planes, so its direction is the cross product of the two normals. For a point on it, set z = 0 and solve the two plane equations for x and y.

The shortest distance from a point P to a plane runs along the normal. Substitute P into ax + by + cz, subtract d, take the size of the result, and divide by |n|. The division converts lengths of n into ordinary distance.

See The intersection of a line and a plane, The intersection of two planes and The distance from a point to a plane.

How do vectors describe motion?

If an object starts at $r_0$r₀ and moves with constant velocity v, its position after t seconds is $r=r_0+vt$r = r₀ + v t.

Two objects collide only if their position vectors are equal at the same time t. Solve one coordinate for t, then test the other.

For the closest approach, subtract one position vector from the other to get the gap as a vector in t. The square of its length is a quadratic in t. Differentiate, set the derivative to 0 to find t, then take the square root for the distance.

When the velocity varies, differentiate the position vector coordinate by coordinate for the velocity, and again for the acceleration. An object that sets off k seconds later has position r(t − k).

See Position and velocity as vectors, Meeting points and travel times, The closest approach of two objects, Velocity that varies with time and Motion that starts later.

The mistakes worth naming

• Adding magnitudes instead of vectors. A move of 3 then a move of 4 at right angles is a move of 5, not 7.
• Getting b − a backwards. The vector from A to B is b − a, destination minus start.
• Using cosine for a line and a plane. The dot product with the normal gives the angle to the normal. The angle to the plane is 90° minus that.
• Assuming crossing paths mean a collision. Two objects can share a route without being at the same point at the same time.
• Skipping the division by the normal's length. Substituting a point into the plane equation gives a distance in lengths of n. Divide by |n|.

Learn this properly in the app

Every lesson linked above is an illustrated screen in Math Challenge with a diagram, a worked example and practice questions. Lines, angles and Pythagoras covers the right triangle behind the magnitude formula, matrices and determinants write the same geometry another way, and triangle trigonometry and bearings solves the same problems without vectors.