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Course: Precalculus>Unit 6

Lesson 5: Vector addition and subtraction

Sal shows how to add vectors by adding their components, then explains the intuition behind adding vectors using a graph.

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• Why do vectors "combine" to form new vectors like that?
• The intuition behind this "combination" is that the resultant vector of ,say, 2 vectors would be the addition of those vectors.
Example : If the displacement of a person is 5 miles east ,and then 2 miles south ,their resultant displacement vector would be the sum of the 2 previous vectors.
• Don't we put the X and Y values in a matrix sort of form? Where the X is above and the Y below?
• I think you are referring to the vector multiplication. Here Sal is talking about addition and subtraction.
• I am stuck on the following:

eg. u (-6, -6) w (-8, -7)
u-w = (-6 (-8), -6 (-7))

I dont know to graphically represent this! any help?
• Instead of thinking it as subtracting w think of it as adding negative w. So negative w is like scaling w by -1 which you probably learnt in one of the previous videos. This makes (-8*-1,-7*-1)=(8,7). So take the vector u and add the vector -w to u. the way to graph it is just graph u from the origin and then graph -w by placing the initial point at the terminal point of u and drawing a line from the initial point of u which is the origin to the terminal point of -w which would be at (2,1). So when subtracting the two vectors, the new vector is equal to a x component of 2 and a y component of 1.
• Hey when do add and subtract vectors? like in what scenarios would you add the vectors or subtract them?
• You would add and subtract vectors if you were trying to plot the direct route to a certain point. Say, Bob went north 9 meters and then went East for 12 meters. 9m @ 90° + 12m @ 0° = 15m @ 36.87°
So you could go 15m at a 36.87° angle to get to Bob "as the Crow flies."
• When adding vectors, do you have to always write it out like how he did it in the video, or could you just be a little quicker and do it in your head? Is there a difference?
• if you are just doing a calculation that you are comfortable with, it makes little sense to write it out, when you can just do it in your head. But if you are not completely adept at the skill, you will make mistakes occasionally, and writing it down lets you check your process to see where/how/why the mistake was made.

You dont learn from doing calculations in your head, you learn from making mistakes, and then figuring out why.
• and why do we also use matrices for vectors operations ?
• Vectors can be seen as nx1 matrices, So vector operations are basically an extension of matrix operations
• I don't understand what Sal means when he says that these vectors are 2 Dimensional (I'm pretty sure he mentioned this in another video). What does 2D mean in this circumstance? What would 3D look like?

• Another way to think about it is the number of ways that a shape can move. A 3D shape can move up/down, left/right, and backwards/forwards (3 ways). A 2D shape can move up/down and left/right (2 ways). A 1D shape can move left/right (1 way). And a 0D shape cannot move at all. The only thing to remember with this way is that a dimension is not restricted to one direction of movement, so a shape that can move in/out (however that works) is in the 1D just as much as a shape that can move left/right.
• At , Sal says '2-dimensional vectors'.
Do 3-dimensional vectors exist?
• Yes. There are vectors in any number of dimensions, including infinite-dimensional vectors.
• Why is it that you can add vectors this way? In triangles, two sides added together must be greater than the remaining side. I understand that you can prove it using a parallelogram method but I am still very confused. Can someone explain this in layman's terms?
• Adding vectors involves combining the direction and magnitude of two vectors to find a resultant vector. This is done using the parallelogram method, which involves drawing the two vectors as sides of a parallelogram and then finding the diagonal that connects the two endpoints. The resultant vector is equal in magnitude and direction to this diagonal.

The reason why this method works is due to the properties of vector addition. Unlike the sides of a triangle, which have a fixed length and cannot be changed, vectors can be translated and shifted in space without affecting their properties. When we add two vectors, we are effectively translating and shifting them so that their endpoints meet, and then finding the vector that connects the initial point of the first vector to the final point of the second vector.

This resultant vector represents the combined effect of both vectors and is equivalent to the diagonal of the parallelogram formed by the two vectors. The parallelogram method works because the opposite sides of a parallelogram are parallel and equal in length, meaning that the magnitude and direction of the diagonal can be easily determined using basic geometry.

In summary, the reason why we can add vectors using the parallelogram method is due to the properties of vector addition and the fact that vectors can be translated and shifted in space without affecting their properties. By drawing the two vectors as sides of a parallelogram, we can find the resultant vector by finding the diagonal that connects the two endpoints.