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What is the difference between scalars and vectors?
A scalar is a quantity that has no direction. It is simply a scale of amounts like kilograms or centimetres. For instance, your weight and height are expressed in terms of an amount and a unit, but they have no direction. Examples of scalar quantities are speed, mass, temperature, energy, length, and distance.
A vector, on the other hand, has magnitude and direction. The momentum of an object, for instance, is equal to its mass per acceleration and has a direction, which makes it a vector unit. Examples of vector quantities are velocity, acceleration, momentum, displacement, and force, including weight.
Resolving vectors into components
Resolving vectors into components helps us when we are dealing with complex vector problems. In order to resolve a vector into its components, we need to measure the horizontal and vertical length of the vector and state these lengths as two separate magnitudes. Let’s take a look at the example below to understand the concept better.
Find the components of the vector shown below.
To find the components of this vector, we need to begin by determining its horizontal and vertical lengths.
As you can see, the horizontal length is 12, and the vertical length is 10. When we resolve a vector into its components, we always get one horizontal value and one vertical one. The lengths we have measured are the magnitudes for the components of the vector.
As you can see, the components of this vector are two vectors, a horizontal one and a vertical one, with magnitudes of 12 and 10.
Can we resolve a vector into its components when we can’t measure its horizontal and vertical lengths? Yes, we can, but let’s take a look at how it’s done.
If we know a vector’s gradient angle, we can determine the magnitude of its horizontal and vertical components. For the vector v above, the gradient angle is a. We can then determine the relationship between the angle and magnitude of the components with the help of trigonometry.
Let’s determine the magnitude of the horizontal component vx. We know that:
If we solve the equation for vx, we get:
Let’s now determine the magnitude of the vertical component vy. Again, we know that:
If we solve the equation for vy, we get:
Adding vectors together
Adding two vectors together is called finding their resultant. There are two ways for adding vectors together. The first involves using scale diagrams, while the second uses trigonometry.
Determining the resultant vectors by using scale diagrams
In order to find the resultant vectors by using scale diagrams, we need to draw a scale diagram of the vectors we wish to add together, connecting the vectors ‘tip-to-tail’.
The following example illustrates the concept.
A man initially walks northeast for 11.40 metres, then continues to walk east for 6.6 metres, and finally walks northwest for 21.26 metres before stopping. Determine the total displacement of the man.
To determine the man’s total displacement, we need to state the lengths he walked as vectors, each with the correct direction and magnitude. Let’s name his first movement as vector A, his second as vector B, and his third as vector C.
If you measure the total displacement with a ruler, you will see that it is 23.094 metres in the northern direction, even though the man walked for 39.26 metres. Let’s prove this mathematically by resolving the vectors into their components. In this particular example, we only need the vertical components since the total displacement is only vertical.
To determine Ay, we apply the equation for resolving vectors into their components:
We don’t have to determine the components of B, as this example doesn’t include a vertical component. For determining Cy, we apply the same equation.
The total displacement is the sum of Ay and Cy, which can be calculated as follows:
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Determining the resultant vectors by using trigonometry
If two vectors are perpendicular to each other, we can find the resultant using trigonometry. Let’s again look at an example.
Two friends are pushing a box. The two forces they apply are perpendicular to each other. One of the friends is applying a force of 3 Newtons (F1) in the eastern direction, while the other is applying a force of 4 Newtons (F2) in the northern direction. Determine the resultant vector for the total force that is being applied to the box.
Two forces, F1 and F2, are perpendicular to each other, which means that the magnitude of Ftotal is equal to the hypotenuse of the triangle formed by these vectors.
Vector Problems - Key takeaways
- Physics uses vectors to express any quantity that has a direction and magnitude.
- To resolve a vector into its components, we need to measure the horizontal and vertical lengths of the vector and express them as two separate vectors.
- To add vectors together, we can use scale diagrams or trigonometry.
- To determine the resultant vectors by using scale diagrams, we need to connect the vectors ‘tip-to-tail’.
- If two vectors are perpendicular to each other, we can find the resultant using the Pythagorean theorem.
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Frequently Asked Questions about Vector Problems
How can we solve vector problems?
To solve vector problems, we apply one of two techniques for adding up vectors. One uses scale diagrams, while the other applies trigonometry. For complex vector problems, it is essential to know how to resolve vectors into their components.
How can we solve a unit vector problem?
In order to solve a unit vector problem, we need to divide the vector by its magnitude.
How can we solve vector word problems?
In order to solve vector word problems, we need to take the given variables and draw a scale diagram, while putting the variables in the right places.
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