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# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.A.1

1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

Sometimes, a number is enough. You know, that dream you have, where all you seem to hear is "Thirty million dollars!" The rest of the dream is a blur, but that one single number is enough to evoke images of private jets, vacations in the Caribbean, and that long-awaited Netflix subscription.

Other times, numbers need a little help. They need direction. That's where vectors come in.

Students should know that vectors are directed line segments, having both magnitude and direction. That means a vector not only tells you how fast the wind is blowing, but also the direction it's blowing in.

They should also know the symbols for vectors. (Mathematicians love symbols and shortcuts, and vectors are no exception.) Instead of calling a vector AB, which is far too much work, mathematicians prefer to call it v (for vector, obviously). In general, boldfaced lowercase letters represent vectors.

If students want to refer to the magnitude, or length, of the vector, they put it inside a double absolute value: ||v||.

Two vectors that have the same magnitude and direction are said to be equivalent (though we use an equal sign to represent equivalency). Equivalent vectors can be thought of as congruent segments on parallel lines: same direction and same length.

The direction of a vector is determined by finding the slope of the segment between its initial point and its terminal point, while the magnitude of a vector is the same as the distance between its endpoints. A vector is said to be in standard form if its initial point is the origin.

#### Drills

1. Is it possible for two vectors to have the same length and not be equivalent?

Yes, if their directions are different

Equivalent vectors need to fulfill two requirements: they must have both the same length and the same direction. If one of these requirements isn't met (or if both aren't met, obviously), then the two aren't equivalent.

2. Which formula would help you find the magnitude of a vector?

Distance

"Magnitude" is just a fancy word for "size." The size of a vector is its length, and length means distance. Slope would only get us direction and the midpoint wouldn't tell us anything useful. While the absolute value of a vector means the same as its magnitude, the term "absolute value" isn't a formula that we can use to help us calculate it.

3. What information is needed to determine whether or not two vectors are equivalent?

Both sets of initial and terminal points

Determining the equivalency of two vectors means knowing their directions and their magnitudes. If we know only the initial points or terminal points of the vectors, their directions and magnitudes are still unknown. Knowing one set of initial and terminal points will give us the direction and magnitude of one of the vectors, but what about the other one? (D) is the only answer that tells us the directions and magnitudes of both vectors.

4. Vector v has an initial point of (0, 2) and a terminal point of (3, 5). What is the vector's slope?

1

Bust out your skis, because it's time to hit some slopes. The slope of a vector is like the slope of a line: rise over run. That means we take the difference of its y coordinates and put it over the difference of its x coordinates. If we do that, we end up with , or just 1. Get ready to catch air off those moguls.

5. Vector v has an initial point of (0, 2) and a terminal point of (3, 5). What is the vector's magnitude?

The magnitude of a vector is its length. To calculate the length of a segment, we just use the distance formula. That means we have ||v|| = . That ends up being , so (C) is our answer.

6. Both vectors n and m have slopes of 2. Are they equivalent?

The magnitude of each vector is needed

Equivalent vectors must have identical slopes and identical magnitudes. Since n and m have the same slope, we need their magnitudes before we can conclusively say that they're equivalent. (We could just say so, but we'd be lying.) That eliminates (A) and (B) automatically. Since we don't have the initial points of the vectors, the terminal points wouldn't help us find their magnitudes. That means (D) is right.

7. Vectors p and q both have magnitudes of . Are they equivalent?

The initial and terminal points of each vector are needed

Vectors have both length (magnitude) and direction. In order to be considered equivalent to each other, both their magnitudes and directions must be the same. We've got magnitude down, but what about direction? Since we don't know their directions yet, (A) and (B) aren't true. We already have the magnitudes, so they aren't needed. That leaves (C) as our only choice.

8. Vector b has a slope of -¾ and a magnitude of 5. If its initial point is (-3, 2), what is its terminal point?

(1, -1)

The easiest way to approach this problem is to plug in and check. While it's possible to do fancy algebraic manipulations with the slope and distance formulas, it might take a while just to solve for one variable. If we plug in each point and check, we'll quickly see whether or not something matches or doesn't. If we do so, the only point that matches both slope and magnitude is (A).

9. Vector v has an initial point at (1, 2) and a terminal point at (4, 4), while vector w has an initial point at (2, 1) and a terminal point at (4, 4). Are the two vectors equivalent?

No, they have the same magnitudes but different directions

We can use the distance formula to calculate the magnitudes of the vectors and the slope formula to calculate the directions. Looking at the magnitudes, we have  for ||v||, compared to  for ||w||. So they both have the same magnitude. The slopes are  for v, and  for w. So they have different directions.

10. Vector g has an initial point (1, 1) and a terminal point of (9, 3). If vector h has an initial point at (-7, 3), what must its terminal point be in order to be equivalent with g?