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Course: Precalculus (Eureka Math/EngageNY)>Unit 2

Lesson 3: Topic D: Lessons 17-18: Vectors in the coordinate plane

Equivalent vectors

Sal determines if two vectors shown on a graph are equivalent by seeing if they have the same magnitude and direction.

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• What does it mean to put a vector inside 2 bars on each side?
(llwll)
• Yes, it means the magnitude or the size of the vector. For example, if you were going in a car and you apply brakes as you see a stop signal, the car will decelerate at a rate. That rate can be -5 meters per second square or anything. This value (-5) is the vector quantity and the magnitude of it would be 5, since the absolute value of -5 is 5. If you're unfamiliar with absolute values, check out khan academy videos on them.

Hope it helped :)
• If you eyeball it, the vectors do seem parallel. But is there a sure-shot method to check if the vectors are parallel? If yes, please specify.
• You can determine if two lines are parallel by finding the slope of both lines. If you find that the slope of each line is equal, then you MAY be parallel. If the slopes are exactly the same, the two lines could either be parallel or right on top of each other. You have to find the y intercept (set x=0 when you have the line in standard form y=ax+b and solve for y) of the two lines. If the y intercepts are different, then the two lines are parallel. This is because the lines will be rising and running (have equal slope) at the same rate, but aren't on top of each other.
• Isn't the slope the magnitude of the vector?
• No. The magnitude of these vectors is found the same way we find the length of the hypotenuse of a right triangle using the Pythagorean theorem. Pretend the vector is the hypotenuse of a right triangle. Break the vector down into its horizontal and vertical components as though they were the legs of that right triangle, square them, add the squares, and take the square root of that sum. That is the magnitude.
The magnitude of the vector is the square root of the sum of the squares of the vertical and horizontal components. The slope is really the direction.
• But how does one say that they are of the same magnitude??

just by the nature the it goes top right?? that says they are both in the sane direction??
• Magnitude is the length of the vector, which Sal finds out in the video using Pythagoras' Theorem. This is also the basis for the Triangle Law of Vector addition. The direction of 2 vectors is the same, if they act *along the same line, OR along 2 parallel lines.*

Assuming A and B to be vectors,
``A = B iff | A | = | B | & direction of A || direction of B.``

(Since 2 lines on top of each other are parallel to each other.)
• Should I Master mathematics in order to study physics in more depth?
• Yes. I am taking physics and calculus at the moment and am amazed at how much that is taken for granted in physics can be explained and can be derived through calculus or other forms of math. Also, I believe that physics is just applied math, and to learn how to apply math, you will need to know how it works.
• How exactly can vectors be equivalent with just the same magnitude and direction (Like at around )? Why don't the location of the vectors (their initial and terminal points) affect whether or not they are equivalent?
• Because a vector is defined by its Magnitude and Direction, but the starting point can be any where, unless it's specified to be the origin, hope that help .
• if two vectors while having same magnitude and direction is that equivalent
• Well not really. Only vectors with the same components can be computed meaning a vector with an X-component can be added with another vector with an X-component also meaning u can't compute vectors with different components as in adding or subtracting an X-component with a Y-component. It is after u find the computation result of X and Y component that u can find the resultant. So unless the components are the same two vectors with unknown components can't be equivalent.
(1 vote)
• What is the difference between "IIuII" and "IuI" ?? How are they equal??
• These are two types of notation to mean the same thing in vectors. More often you will see ||u|| in vectors for the magnitude, and |u| for a matrix u to talk about it's determinant. Since they are very much related, the notation occasionally crosses over from one to the other.
The magnitude is equal since the they have the same length. (use the distance formula, or Pythagorean theorem).
• At , what does the triangle beside y indicate? Is this symbol used for other things too?
(1 vote)
• You're right, this symbol is in fact a very important symbol in most of mathematics. It is the Greek letter delta (mostly Δ, sometimes 𝛿) which, at least in this case, means 'change in'. Think of it as just an important fancy symbol because as you go further into calculus, this symbol will keep showing up.

In the video, Sal writes 'Δy' (read as delta y) instead of 'change in y' because it is the more formal notation. (just like the arrow above vectors)