Introduction to Vectors What is a vector? Algebra of vectors The scalar product

What is a Scalar? Scalars are quantities that are simply defined by a magnitude. For example the mass of an object just has a magnitude, so it is a scalar quantity. Another example of a scalar is speed!

Without Scalars We Wouldn’t Be Able To Model Temperature. Mass of an object.

What is a Vector? A vector quantity has both a direction and a magnitude (size). For example velocity is a speed in some given direction, for example wind. The wind is blowing 8 mph in a westerly direction.

Without Vectors We Wouldn’t Be Able To Model Torque and circular motion. The forces acting on an aircraft.

Scalar or Vector? Which of the following are vector quantities?

Intended Learning Outcomes By the end of this section it is intended that you will be able to: Define what is meant by a vector and differentiate between a vector and scalar quantity. Algebraically manipulate a vector. Calculate the angle between two vectors using the scalar product.

Properties of Vectors Cartesian Vectors Scalar Product

Properties of Vectors

Representation of Vectors A B a

Modulus of a Vector The modulus or magnitude of a vector is written in the form |a|. For example, if we had the vector a, we would write its magnitude as |a|.

Equality of Vectors

Negative Vectors ab

Multiplication by a Scalar If we have some real positive constant λ, and multiply a vector a by it, i.e. we have λa then we have simply changed the length of our by an amount λ. For example if we multiplied a by 2, the length of the vector would double! a 2a

Multiplication by a Scalar If we multiply by a number less than one, we shorten the length of our vector. For example if we multiplied a by 0.5, the length of the vector would be halved! If we were to multiply by a negative number, we would reverse the direction of the vector. a 0.5a

Unit Vectors

Angle Between Two Vectors If we consider two vectors, there are two possible angles between them, φ and 180 – φ. The angle two vectors is uniquely defined. It is the angle between their directions when the lines representing them diverge φ φ φ φ

Cartesian Vectors

To describe the point in space where an object lies requires the use of a co-ordinate system. The co-ordinate system we use is the cartesian co-ordinate system. We first define a unit vector in each of the x-, y- and z- directions.

Cartesian Vectors i is the unit vector along the x-axis. j is the unit vector along the y-axis. k is the unit vector along the z-axis. y x z j i k

Vectors in 2D Vector Form r = i + j Cartesian Form

Draw the Following Vector Draw the following vector r = 2i + 3j To do so, we must first move 2 units along the x-axis, then 3 units along the y-axis. y x 2 3

Vectors in 3D

Draw the Following Vector Draw the following vector r = 2i + 3j + 2k To do so, we must first move 2 units along the x-axis, then 3 units along the y-axis and finally 2 units along the z-axis. y x z 2 3 2

Algebra of Vectors We should now move on to the algebra of vectors: Addition Multiplication by a scalar Magnitude of a Vector

Addition of Vectors Vector 1Vector 2 r 1 = r 2 = r 1 + r 2 =

Multiplication by a Scalar r =

Magnitude of a Vector y x z axax azaz ayay Calculate the magnitude of the vector a = (a x, a y, a z ). We should first find the magnitude of the purple line (w), by first considering the xz plane. |a||a| w

Magnitude of a Vector x z azaz axax w

y ayay |a||a|

Example

The Scalar Product Otherwise known as the inner product or dot product.

The Dot Product

Calculate the Angle Between Two Vectors To calculate the angle between two vectors we need to use the dot product. Let us begin by considering the following triangle. a b b-a φ

Calculate the Angle Between Two Vectors a b b-a φ

Dot Product – Try it Yourself Vector 1Vector 2 r 1 = r 2 = Angle between vectors =

Example of Dot Product

Example of the Dot Product

Properties of the Dot Product Parallel Vectors: If we have two vectors a and b which are parallel to each other, i.e. a ab b a.b = abcos(0) = |a||b|a.b = abcos(π) = -|a||b| For like vectors a.b = |a||b|For unlike vectors a.b = -|a||b|

Properties of the Dot Product Perpendicular Vectors: If we have two vectors a and b which are perpendicular to each other, i.e. The angle between them is 90 o, and we see that a.b = 0. If two vectors are perpendicular, their dot product is zero. a b

Conclusion Now that you are at the end of this section, you should be able to: Define what is meant by a vector and differentiate between a vector and scalar quantity. Algebraically manipulate a vector. Calculate the angle between two vectors using the scalar product.