Chapter 3: Vectors

Question 1

Week 1

What is a vector?

Answer 1

A vector is a quantity that contains two pieces of information:

• A magnitude.
• A direction.

Question 2

Week 1

What is the difference between a scalar and a vector?

Answer 2

A scalar is quantity that only contains one piece of information: a magnitude.

A vector is a quantity that contains two pieces of information: a magnitude and a direction.

Question 3

Week 1

What are tensors?

Answer 3

Tensors are a family of quantities that contain information.

Scalars and vectors are both a member of the tensor family; they are first-order tensors.

Question 4

Week 1

Name the simplest vector quantity.

Answer 4

The displacement vector is a very simple vector quantity.

All a displacement vector really is distance traveled in a specific direction.

For example, let’s say that a person travels from point A to B in a straight line. That person’s displacement and distance has the same magnitude, but the displacement quantity will have an extra piece of information: the direction.

Hence, the person’s displacement is x meters in the direction of B.

Question 5

Week 1

How do you add vectors using the Head to Tail method?

Answer 5

For a detailed answer, watch this YouTube video.

Question 6

Week 1

How do you split a vector into its components?

Answer 6

Each vector quantity has two pieces of information its magnitude and its direction

Here’s how you determine both sets of information from the vector’s co-ordinates.

Suppose we have a two dimensional vector with co-ordinates ai + bj

• Here’s how we determine the magnitude of the vector.

√(a)² + (b)²

• Here’s how we determine the direction of the vector with respect to the positive x-axis

tan^-1 (b/a)

Question 7

Week 1

Name the different types of vectors.

Answer 7

• Displacement vectors
• Unit vectors
• Position vectors
• Velocity vector
• Torque vector

These are just a few examples out of the many that exist.

Question 8

Week 1

What is a position vector?

Answer 8

A position vector indicates either the position or the location of any given point with respect to any arbitrary reference point like the origin.

Suppose that you are walking to the nearest tree from your house. You walk three steps to the right, and walk five steps forward to get to the tree.

Your position vector with respect to the house will be 3i + 5j

The house in this example is an arbitrary point that you chose to tether yourself to, just like the origin. Now, wherever you go, your position will be represented in terms of how far you are from your house.

Question 9

Week 1

What is a unit vector?

Answer 9

A unit vector, just like any vector, has a magnitude and a direction, but with one caveat.

A unit vector has a magnitude of 1, but it retains its directional properties.

How do we calculate it?

The unit vector of a vector is calculated using the following formula.

unit vector = vector/magnitude of vector

Question 10

Week 1

What is the dot product of a vector?

Answer 10

We can multiply vectors in two different ways.

The Dot Product (Scalar Product)

The Dot Product is a vector multiplication operation that results in a scalar product.

Here are some of the properties of the dot product.

• The dot product of two perpendicular vectors is 0.
• The dot product of two vectors is commutative i.e. a . b = b . a
• The dot product of two vectors a and b is |a||b|cosθ
• The dot product of a vector and itself is equal to the magnitude of said vector squared.
• The dot product follows the associative law.

How Do We Calculate the Dot Product?

The dot product of two vectors (a b c) & (d e f) is a x d + b x e + c x f

Question 11

Week 1

What is the cross product of a vector?

Answer 11

The Cross Product (Vector Product)

The cross product is one of two ways of multilpying two vectors.

The cross product is a vector multiplication operation that results in a vector product.

Here are some properties of the cross product.

• The cross product of two vectors is not commutative i.e. a x b ≠ b x a.
• The cross product of two vectors is not associatve i.e (a x b) x c ≠ a x (b x c).
• The cross product of two vectors is distributive.
• The cross product of two vectors a and b is |a x b| = |a||b|sinθ.
• The vector product of two vectors a and b is perpendicular to the plane containing both vectors.
• You can use the right hand rule to find the direction of the cross product given that you know the direction of the vectors being multiplied.

Question 12

Week 2

What is the origin?

Answer 12

The origin is a hypothetical point that does not exist i.e. it is not quantifiable and is the absence of magnitude.

We just find it convenient to relate vectors (namely position vectors) to an arbitrary point because it makes things a lot easier.

Question 13

Week 1

Is time a dimension?

Answer 13

Time is not a part of our spatial dimension.

What constitutes a spatial dimension?

In a spatial dimension, an object (or a point mass) can be moved around. In our world, three spatial dimensions exist (3-D).

Now time, per se, does not exist in out spatial dimension since it does not follow its tenets i.e it cannot be moved around. Time is linear, and does not have a specific origin.

However, our universe would be very different without time. Think about it: our lives would start and end instantaneously.

Therefore, time exists in its own dimension, known as the temporal dimension.

Together, the spatial and temporal dimensions. (spacetime) essentially describe how our universe functions.