Introduction to Vectors - Online Tutor, Practice Problems & Exam Prep

Understanding vector math is crucial in physics, particularly when combining vectors with direction. When adding parallel vectors, such as 3 meters right and 4 meters right, simply sum them to get 7 meters. However, for perpendicular vectors, like 3 meters right and 4 meters up, use the Pythagorean theorem: a2+b2=c2, leading to a resultant displacement of 5 meters. Mastering these concepts is essential for solving complex physics problems.

Introduction to Vector Math

Hey, guys. We're going to be working with vectors a lot in physics, so you're going to have to get very good at how we manipulate and combine them. In this video, I want to introduce to you what vector math is and what it's all about. But I want to take a minute to describe what the next few videos are going to be about. We're going to be working with vectors, with these diagrams, with these little axes and grids, and boxes because it's a great way to understand very visually what's going on with vectors. Then later on, what we're going to see is we're going to see all of the math equations that describe vectors. So let's get to it.

Adding and subtracting scalars is pretty easy. Remember, scalars are just simple numbers. But vectors have directions, so math is sometimes not as straightforward. For example, if you were to combine scalars, let's say you're moving and you're combining a 3-kilogram and 4-kilogram box, then the way that you add these boxes together if you were to lump them together in a single box is you just add 3 and 4 straight up. So 3+4 is just 7. Combining scalars is just simple addition.

Where things get a little bit more tricky is when you start combining vectors, and there are really just 2 cases that you're going to see. You combine parallel vectors, then there would be like walking 3 meters to the right and 4 meters to the right. So, because vectors have direction, they're drawn as arrows. This 3 meters to the right and 4 meters to the right would look like this. They would just add together because they are both in the same direction, resulting in a total displacement of 7 meters.

Where things get a little trickier is when you're combining perpendicular vectors. So now let's say I walk 3 meters to the right and then 4 meters up. So we got this grid that's going to help us visualize what's going on here. Now, you might think that the total displacement is just 3+4, which is 7, but it's actually not because your total displacement is really just the shortest path from where you start to where you end. What happens is we make a little triangle like this. The way we solve for this is by using an old idea, an old math equation from algebra or trigonometry called the Pythagorean theorem. So this is the Pythagorean theorem: c 2 = a 2 + b 2 . This means my total displacement is 3 2 + 4 2 , which is actually equal to 5 meters.

We're going to continue with more examples. For instance, if we walk 10 meters to the right and then 6 meters to the left. The total displacement, after drawing the displacement vectors, is just 10 minus 6, which is 4 meters, because these displacements are antiparallel but along the same line. Now, for part B, if I walk 6 meters to the right and then 8 meters down, we have perpendicular vectors again, forming another triangle. The displacement, or the hypotenuse of this triangle, is solved using the Pythagorean theorem: c 2 = 6 2 + 8 2 , resulting in a displacement of 10 meters.

Alright, guys. That's it for this one. Let me know if you have any questions.

Two perpendicular forces act on a box, one pushing to the right and one pushing up. An instrument tells you the magnitude of the total force is 13N. You measure the force pushing to the right is 12N. Calculate the force pushing up.

• A. 1 newton
• B. 5 newtons
• C. 12 newtons
• D. 18 newtons

Here’s what students ask on this topic:

What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude, such as mass or temperature. It is represented by a single number. A vector, on the other hand, has both magnitude and direction. Examples include displacement, velocity, and force. Vectors are often represented as arrows, where the length of the arrow indicates the magnitude and the direction of the arrow indicates the direction of the vector.

How do you add parallel vectors?

Adding parallel vectors is straightforward. If the vectors are in the same direction, you simply add their magnitudes. For example, if you have two vectors, 3 meters to the right and 4 meters to the right, the resultant vector is 7 meters to the right. If the vectors are in opposite directions, you subtract the smaller magnitude from the larger one. For instance, 10 meters to the right and 6 meters to the left result in a vector of 4 meters to the right.

How do you add perpendicular vectors?

To add perpendicular vectors, you use the Pythagorean theorem. For example, if you walk 3 meters to the right and then 4 meters up, you form a right triangle. The resultant vector is the hypotenuse of this triangle. Using the Pythagorean theorem: $a^2+b^2=c^2$, where $a$ is 3 meters and $b$ is 4 meters. The resultant vector $c$ is $\sqrt{3^2+4^2}=\; 5$ meters.

What is the Pythagorean theorem and how is it used in vector math?

The Pythagorean theorem is a fundamental principle in geometry that states: $a^2+b^2=c^2$, where $a$ and $b$ are the lengths of the legs of a right triangle, and $c$ is the length of the hypotenuse. In vector math, it is used to find the resultant vector when adding perpendicular vectors. For example, if you have vectors of 3 meters and 4 meters at right angles, the resultant vector is $\sqrt{3^2+4^2}=\; 5$ meters.

How do you calculate the resultant displacement of two vectors?

To calculate the resultant displacement of two vectors, you need to consider their directions. For parallel vectors, simply add or subtract their magnitudes. For perpendicular vectors, use the Pythagorean theorem. For example, if you walk 6 meters to the right and then 8 meters down, you form a right triangle. The resultant displacement is the hypotenuse, calculated as: $\sqrt{6^2+8^2}=\; 10$ meters.