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Vectors

Why this chapter

Why this chapter exists

A displacement, a velocity, a force and a field each have a size and a direction. They do not add as plain numbers: a pull of \(3\,\text{N}\) and a pull of \(4\,\text{N}\) may give \(1\,\text{N}\), \(5\,\text{N}\) or \(7\,\text{N}\). This chapter gives the rules for such quantities. It adds them tip to tail, splits them into components, and multiplies them in the two ways that physics needs, for work and for torque. Every chapter from Motion in Two Dimensions onwards uses these rules on almost every page.

You will learn

By the end of this chapter you will

Tell a scalar from a vector, and scale, negate and compare vectors.
Add and subtract vectors tip to tail, and find a resultant with \(R^2 = A^2 + B^2 + 2AB\cos\theta\).
Resolve a vector into components along any pair of perpendicular axes, and add many vectors by their components.
Find a dot product and a cross product from an angle or from components, and use them for work, flux, torque and magnetic force.
Use position vectors and relative vectors to find a displacement, a closest approach and a centre of mass.

Before you begin

You should be able to use the theorem of Pythagoras and read a coordinate on a graph. You should know the sine, cosine and tangent of an angle in a right triangle, and the values at \(30°\), \(37°\), \(45°\), \(53°\) and \(60°\). The Toolbox chapter on trigonometry covers them. No vectors are assumed.

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Lessons

Scalars and Vectors

Quantities that add as plain numbers and quantities that also need a direction. Magnitude, equal and negative vectors, the zero vector, and a vector multiplied by a number.

Scalars and vectorsEqual negative and zero vectorsMultiplying a vector by a number

14 min0%

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Adding Vectors Tip to Tail

The triangle, parallelogram and polygon laws, the magnitude and direction of a resultant by \(R^2 = A^2 + B^2 + 2AB\cos\theta\), and subtraction as the addition of a negative vector.

Adding vectors tip to tailThe parallelogram and the polygonSubtracting a vector

16 min0%

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Components and Unit Vectors

Splitting a vector into perpendicular components, \(A_x = A\cos\theta\) and \(A_y = A\sin\theta\), the unit vectors \(\hat i\), \(\hat j\), \(\hat k\), going back to a magnitude and a direction, and adding vectors by their components.

Components and unit vectorsMagnitude and direction from componentsAdding vectors by components

16 min0%

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The Dot Product

The product of two vectors that is a number, \(\vec A\cdot\vec B = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z\). The projection of one vector on another, the angle between two vectors, the test for perpendicular vectors, and work.

The dot productThe dot product from componentsProjection and the perpendicular test

15 min0%

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The Cross Product

The product of two vectors that is a vector, \(|\vec A\times\vec B| = AB\sin\theta\), at right angles to both by the right-hand rule. The area it measures, why the order matters, and its components in the determinant form.

The cross productThe cycle of unit vectorsThe cross product from components

16 min0%

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Relative and Position Vectors

The position vector, the displacement \(\vec r_2 - \vec r_1\), the position and velocity of one body relative to another, the vector equations of motion, the closest approach of two bodies, and the centre of mass as a weighted sum of position vectors.

Relative and position vectorsVector equations of motionThe weighted sum of position vectors

15 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

component: The signed length of the projection of a vector on an axis, such as \(A_x = A\cos\theta\). The components add back to the vector.
cross product: The vector \(\vec A\times\vec B\) of magnitude \(AB\sin\theta\), at right angles to both vectors by the right-hand rule, also called the vector product. Torque is \(\vec r\times\vec F\).
dot product: The scalar \(\vec A\cdot\vec B = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z\), also called the scalar product. Work is the dot product \(\vec F\cdot\vec d\).
magnitude: The size of a vector, written \(|\vec A|\) or \(A\). A magnitude is never negative and carries the unit of the quantity.
position vector: The vector \(\vec r = x\hat i + y\hat j + z\hat k\) from a chosen origin to a point. Its components are the coordinates of the point.
projection: The shadow of one vector on the line of another, the part of the first vector that lies along the second. Its signed length is \(A\cos\theta\).
relative position: The position of one body seen from another, \(\vec r_{AB} = \vec r_A - \vec r_B\), the arrow from B to A.
resultant: The single vector equal to the sum of two or more vectors. Drawn tip to tail, it runs from the first tail to the last tip.
scalar: A physical quantity that has magnitude only, such as mass, time or energy. Scalars of one kind add as ordinary numbers.
unit vector: A vector of magnitude 1 and no unit that only gives a direction. \(\hat A = \vec A/A\) points along \(\vec A\); \(\hat i\), \(\hat j\), \(\hat k\) point along the axes.
vector: A physical quantity that has magnitude and direction and that adds by the triangle law, such as displacement, velocity or force.
zero vector: The vector of magnitude zero, \(\vec 0\), such as the sum of a vector and its negative or the displacement of a closed trip.