Vectors Part 1: Two Dimensions Last modified: 20/02/2018
Links Scalars Vectors Definition Notation Polar Form Compass Directions Basic Vector Maths Multiply a Vector by a Scalar Unit Vectors Example Vectors Adding Vectors Vector Components Resolving Vectors Cartesian Form of a Vector Convert Polar Form to Cartesian Form Convert Cartesian Form to Polar Form Calculate Unit Vector Subtracting Vector Zero Vector Scalar (Dot) Product Definition Mathematical Properties Example Cartesian Vectors Application: Finding the Angle Between 2 Vectors
Scalars Many quantities measured and studied in Physics are completely described by their (a) value and (b) unit. For example, the mass of a particular object might be 5 kilograms. Here the value: 5, and the unit: kilograms tell us everything we need to know. Mass is an example of a scalar quantity (or just scalar). Other scalars include: energy, temperature, volume, density, pressure, power. The mathematics of scalars is the same as for normal numbers. For example, if we have 2 masses, m 1 = 5 kg and m 2 = 3 kg, then the total mass is of course: m total = m 1 + m 2 = 5 kg + 3 kg = 8 kg
Vectors Not every Physics quantity is a scalar. To fully describe some important things we need not just (a) a value and (b) a unit, but also (c) a direction. These are called vector quantities. An example, used to describe motion, is velocity. Two cars approach each other, both travelling at a speed of 50 km/hr: 50 km/hr 50 km/hr Though the cars have the same speed, their motions are different - they have different directions. The cars have different velocity vectors.
50 km/hr The velocity of the blue car is 50 km/hr to the right, and the 50 km/hr velocity of the red car is 50 km/hr to the left. It is usual, as seen here, to represent a vector by an arrow pointing in the appropriate direction. The magnitude (also called the size, length or modulus) of these two vectors (i.e. 50 km/hr) is the same, only the direction is different. Most vectors in Physics, like velocity, have units. For simplicity, while looking at the properties of vectors in this lecture, we will ignore units. Of course we will need to include them when we start doing proper calculations.
Notation Vectors are widely used in different fields of science, engineering and maths. Because of this wide use, there is some variation in the notations used. Particularly in printed material, a vector is usually represented by a letter in bold font, e.g. v. This will be the notation used in these lectures and in tutorials. Of course, it is difficult to use bold fonts in handwriting, and several notations are commonly used in this case: ṽ and v are both popular - either one is acceptable in our exams. The magnitude of a vector can be represented using the absolute value, or simply by the letter in normal font, without an arrow or twiddle : v = ṽ = v = v Be careful: if you mean the vector, include the arrow or twiddle.
Polar Form Describing a vector by giving its magnitude r, and angle θ to the x-axis is called the polar form of the vector. This can be written as an ordered pair: (r, θ). y 4 150 5 40 x The blue vector has r = 5 and θ = 40 (r, θ) = (5, 40 ). Note that the angle is, by default, measured in an anti-clockwise direction. The red vector is (4, 150 )
When the vector is in the third or fourth quadrant, there are two possibilities used to describe the angle: y The angle is 220 in the anti-clockwise direction 220 5 140 x OR 140 in the opposite (i.e. clockwise) direction. We indicate this opposite direction with a minus sign. The vector in polar form is: (5, 220 ) OR (5, 140 ). Both are correct. Many people prefer to avoid using angles greater than 180, so the second form is probably seen more often. Remember: anti-clockwise positive θ and clockwise negative θ.
Compass Directions Especially when dealing with with velocities of cars, boats etc., it is common to express a vector s direction in terms of compass points, rather than x and y axes. This is really quite simple, but may be a little confusing at first. How can we describe the direction of the vector below? N W E 60 S
First, decide which of the main compass directions is closest. In this example: west. W 60 Next, calculate the angle to this nearest direction. Here that is 30. W 30 We could describe the vector as being 30 away from west, but this description fits two different vectors! One on the south side and the other on the north. Our vector is the one 30 on the south side of west. This is usually shortened to: 30 S of W Other directions are described similarly. 30 W 30
Multiply a Vector by a Scalar Multiplying a vector v by a scalar c gives a new vector u: u = cv The direction of u is the same as that of v. The scalar just multiplies the magnitude of v. i.e. u = cv. v 2v 3v 2v 1 2 v Notice that for a negative scalar, the direction of the result is reversed.
Unit Vectors A unit vector is a vector with magnitude = 1. Unit vectors are used to indicate direction. The unit vector in the same direction as a vector v is written as ˆv (usually said as v hat ). Any vector v can therefore be written as the product of its magnitude (v) and its direction (ˆv): v = v ˆv So, if we are given a vector v, the unit vector ˆv can be calculated by: ˆv = v v
Example Vectors When dealing with more than one vector, it is important to make sure that we are using the same scale for all vectors. Drawing vectors on a grid helps with this: w v u These three (randomly chosen) vectors will be used in the following pages to demonstrate some mathematical operations involving vectors.
Adding Vectors The sum of two vectors is also a vector: s = a + b The process to determine s is shown here using our example vectors u and v - the same method can be used to add any two vectors. From the origin, draw vector u then, draw vector v, starting from the tip of u Draw a new vector, from the start of u to the tip of v. This new vector is u + v u + v u v
What if we switch the order and calculate v + u instead? Follow the same procedure: Draw vector v Draw vector u, starting from the tip of v Draw a new vector, from the start of v to the tip of u. This new vector is v + u The result is the same as before: u + v = v + u v u u + v + = uv + u u v Like the familiar adding of numbers, vector addition is commutative (i.e. the order doesn t matter).
If we have carefully drawn our vectors to scale, then determining the properties of the vector s = u + v is straightforward: Draw the two sides of a right triangle, in the x and y directions, with s as the hypotenuse. then, using the grid, find the lengths of these. Use Pythagoras to find the length s = 2 2 + 5 2. 29 From the triangle, θ s = tan 1 (5/2) = 68 2 68 5 The vector s is ( 29, 68 ) or better: (5.39, 68 ).
This definition can be extended to any number of vectors. For example, to calculate the sum of our three example vectors u + v + w: Draw vector u Draw vector v, starting from the tip of u Draw vector w, starting from the tip of v Draw a new vector, from the start of u to the tip of w. This new vector is u + v + w u + v + w w u v Adding 4 vectors, 5 vectors etc is done the same way.
Components of a Vector Let s look again at the addition of two vectors: c = a + b We saw earlier that if we have the lengths c x and c y we can use basic trigonometry to obtain the length and direction of the vector c. c b c y a c x In our example, we were able to easily obtain the lengths c x and c y from our diagram, because in that case, all vectors lined up perfectly with the grid. This won t always happen. How can we easily add two vectors without trying to read numbers off a graph?
We can write the vector c as the sum of two vectors exactly in the x and y directions: c = c x + c y These vectors are called the components of c. Similarly for a and b: c b c y c b b y a c x a a x a y b x From the diagram, it should be clear that: c = c x + c y = (a x + b x ) + (a y + b y ) To find c x,y we need to determine a x,y and b x,y.
Resolving Vectors Calculating the components of a vector is known as resolving the vector into its components. The components of a 2D vector are always in the x or y direction. We need a neat way to indicate these directions. Remember, directions are indicated by unit vectors. Two standard unit vectors are defined in the direction of the axes: i x-direction j y-direction For example, a vector of length 2 along the x-axis is written: 2i.
Cartesian Form of a Vector Let s calculate the components of our example vector u : Draw in the two short sides of the triangle Note, there are two equivalent ways to do this! Express each of these sides as a vector, using i and j 3 4 u (4,3) 3j u = 4i + 3j 4i4 The Cartesian co-ordinates of the end-point of the vector are the same as the magnitudes of the components, so the vector expressed in terms of i and j is called the Cartesian form of the vector.
The same process is followed to find the example vector v in Cartesian form: v = 2i + 2j 2j 2 v 2i 2 Our other example vector in Cartesian form is w = 3i 4j, (Check for yourself!) Once we have two vectors expressed in Cartesian form, adding them together is simple: s = u + v = (4i + 3j) + ( 2i + 2j) = 2i + 5j Generally, if you need to add two or more vectors, you should first express them in Cartesian form.
Convert Polar Form to Cartesian Form For a general vector a in polar form (r, θ), the Cartesian form will be: r r sin θj a = r cos θi + r sin θj θ r cos θi This formula will work correctly for all values of θ. Notice that the unit vector â can be easily calculated: â = a r = cos θi + sin θj Any 2D unit vector can be written in this form.
Convert Cartesian Form to Polar Form Converting a general vector a in Cartesian form a x i + a y j to polar form can be a little bit tricky: If the vector is in the first quadrant, there is no problem: r a y for the example vector u = 4i+3j θ a x Clearly: r = a 2 x + a 2 y and θ = tan 1 ( a y a x ) 32 + 4 2 = 5 θ = tan 1 3 4 37 4i u is 5 at 37 3j
But for a vector in the second quadrant (i.e. a x < 0): a y r a x φ θ = 180 φ Again: r = a 2 x + a 2 y, BUT if we use the same formula as before: θ = tan 1 ( ay a x ), most calculators will give an answer in the fourth quadrant. The correct method is to first calculate φ = tan 1 ) ( ay a x and then find θ = 180 φ. Similar issues will occur in the third and fourth quadrants. Drawing a diagram is really necessary to find the angle correctly.
for the example vector w = 3i 4j 3i φ = tan 1 4 3 53 θ = (180 53) = 127 4j 32 + 4 2 = 5 w is 5 at 127 (or 233 if you prefer) Which form is better - Polar or Cartesian? Neither! Both forms are equally valid ways of describing a vector. Unless an exam question specifically asks you to give your answer in one form, then either is acceptable for your answer. Use whichever is more convenient.
Calculate Unit Vector Finding the unit vector in the same direction as a Cartesian vector is straightforward. Recall that the unit vector â in the direction of a vector a is given by: â = a/a For the example vector u = 4i + 3j : We have already calculated the length u = 3 2 + 4 2 = 5 Therefore û = 1 5 (4i + 3j) = 0.8i + 0.6j û u 3 This unit vector û has length 1 and is in the same direction as u. 4
Subtracting Vectors To calculate the difference between two vectors, we need to realize that subtraction is a special case of addition: d = a b = a + ( b) Using the same vectors as before, draw vector u Draw vector v, starting from the tip of u Draw a new vector, from the start of u to the tip of v. This new vector is u v u u v v
Zero Vector If we subtract a vector from itself, the result is the zero or null vector: 0. a a = 0 = 0i + 0j It is quite common (though technically incorrect) to see 0 written instead of 0 (i.e. no bold font), but the meaning should be clear either way.
Scalar Product The scalar product of two vectors a and b is: a b = a b cos θ = ab cos θ where θ is the angle between the two vectors. As the name suggests, a b is a scalar! b θ a Because of the symbol used, the scalar product is also known as the dot product. Mathematicians often call it the inner product. You shouldn t get too stressed about what what the scalar product means. Think of it as just a mathematical definition that turns out to be useful.
The scalar product multiplies the lengths of the parts of the vectors that lie in the same direction. i.e. a times the component of b in the direcion of a: a b = a(b cos θ) = ab cos θ b θ b cos θ b sin θ a Or equivalently, b times the component of a in the direcion of b: b a b = (a cos θ)b = ab cos θ a cos θ θ a sin θ a
Scalar Product: Mathematical Properties a b = ab cos θ Some useful properties can be deduced directly from the definition: The scalar product is commutative: a b = b a The scalar product is distributive: a (b + c) = a b + a c The product of a vector with itself: a a = a 2 cos 0 = a 2 The product of two perpendicular vectors: a b = ab cos 90 = 0
Scalar Product: Example Calculate the scalar product of the example vectors u and w: y u 5 5 37 x 127 127 + 37 = 164 w u w = 5 5 cos(164 ) = 24.0 Notice that the value of the scalar product can be positive or as in this case, negative.
Scalar Product: Cartesian Vectors Calculating the scalar product of two vectors given in polar form is easy. But what about if we are given two vectors in Cartesian form? For our examples, we have already determined: what is u w? u = 4i + 3j and w = 3i 4j We could use the polar form and the scalar product definition as we just did, but if we only know the Cartesian form it is much quicker to use the distributive property of the scalar product noted earlier. u v = (4i + 3j) ( 3i 4j) = (4)( 3)i i + (4)( 4)i j + (3)( 3)j i + (3)( 4)j j = (4)( 3) + (3)( 4) = 12 12 = 24 where we have also used i i = j j = 1 and i j = j i = 0
For any two vectors a = a x i + a y j and b = b x i + b y j : a b = a x b x + a y b y Remember, from the definition, a a = a 2, so we have: a = a a = ax 2 + ay 2 which is, of course, a rewritten version of Pythagoras theorem.
Finding the Angle Between 2 Vectors How can we calculate the angle between two vectors given in Cartesian form? For example, our friends u = 4i + 3j and w = 3i 4j? One method is to convert both to polar form, then subtract the two angles, as we did a few pages back: 37 ( 127 ) = 164 Alternatively, we can use the scalar product. Rearranging the definition gives: cos θ = u w uw 4 ( 3) + 3 ( 4) = 42 + 3 2 ( 3) 2 + ( 4) = 24 2 25 = 0.96 θ = 164 If we don t already have the vectors in polar form, the second method is much easier.
Example Using the two vectors A and B shown at right, calculate: (a) A + B in polar form. (b) the angle between A and B. (c) the vector C which has the same length as A and the same direction as B. y 5 60 A x B = 2i 3j (a) To add vectors, we need to express them in Cartesian form. B is already in this form, but we need to convert A: A = 5 cos 60i + 5 sin 60j = 2.50i + 4.33j y 5 60 x 5 cos 60 5 sin 60
Adding the vectors is now easy: y A + B = (2.50i + 4.33j) + (2i 3j) = 4.50i + 1.33j In polar form: A + B is (4.7, 16 ). 4.52 + 1.33 2 = 4.7 θ 4.5i 1.33j tan θ = 1.33 4.5 θ = 16 x (b) Finding the angle uses the method seen earlier with the scalar product: A B = (2.50i+4.33j) (2i 3j) = (2.5)(2)+(4.33)( 3) = 8 cos θ = A B AB = 8 (5)( = 0.44 θ = 2 2 + ( 3) 2 116 )
(c) Remember that the direction of vector B is indicated by the unit vector ˆB: ˆB = B B = 2i 3j = 0.55i 0.83j 22 + ( 3) 2 We know that the length of vector A is A = 5 so the required vector C, with the length of A and direction of B, must be: C = A ˆB = (5) (0.55i 0.83j) = 2.8i 4.2j