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Common Core Standards: Math

High School: Number and Quantity

Vector and Matrix Quantities HSN-VM.A.2

2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

As we already know, a vector has both direction and magnitude—position and length. Think of the pitcher's mound as the origin. The pitcher can throw in lots of different directions and lots of different distances depending on where the runner is.

The easiest form of a vector is when its initial point is at the origin. Students should know that a vector like this is said to be in standard form. It's standard because those zeros are very easy to work with. (Unless you put them in the denominators of fractions. That's a sin.)

The components of a vector are the differences in the x and y coordinates of its terminal and initial points.

For example, let's say we have a vector v with initial point (2, 3) and terminal point (3, 5). Its components are the difference in the x values (3 – 2 = 1) and the difference in the y values (5 ­– 3 = 2). That means we could write our vector in component form as v = <1, 2>. 

The two notations are equivalent, since a vector from (2, 3) to (3, 5) is equivalent (by definition, remember?) to a standard vector that starts at the origin and ends at (1, 2). We give these vectors the components <1, 2>, and we use the < and > symbols instead of parenthesis to distinguish them from looking like points.

Drills

  1. Find the components of a vector in standard position that terminates at (-5, 4).

    Correct Answer:

    <-5, 4>

    Answer Explanation:

    News flash: We do have the initial point. The fact that it's in standard position means the initial point is the origin. To get the first component, subtract the x values to get -5 – 0 = -5. To get the second component, subtract the y values: 4 – 0 = 4. Vectors in standard position are the quick ones!


  2. Find the components of a vector with initial point (-4, 5) and terminal point (4, 5).

    Correct Answer:

    <8, 0>

    Answer Explanation:

    To find the first component, subtract the x values. That gives us 4 – (-4) = 8. To find the second component, subtract the y values, which equals 5 – 5 = 0. That means the vector's components are <8, 0>, or (B).


  3. If we switch the initial and terminal points of a vector, will its components change?

    Correct Answer:

    Yes. Both x and y components will change sign.

    Answer Explanation:

    If we subtract the numbers in the reverse order, both components will change signs. We can try this with an initial point of (-1, -1) and a terminal point of (1, 1). If we subtract them as is, we'll get <2, 2> for its components. If we switch them, we'll get <-2, -2>. That sounds like (B) to us.


  4. Find the components of a vector in standard position with terminal point (9, 0).

    Correct Answer:

    <9, 0>

    Answer Explanation:

    Since the vector is in standard position, simply subtract the coordinates of origin from the coordinates of the terminal point. In other words, the terminal point of a vector in standard position is the same as its components. So a vector in standard position with terminal point (9, 0) has the components <9, 0>. Easy peasy.


  5. A vector in standard position has a terminal point of (-3, 4). What are its components?

    Correct Answer:

    <-3, 4>

    Answer Explanation:

    To find the components of a vector, we subtract its terminal x and y coordinates from its initial x and y coordinates. Since the initial coordinate for a vector in standard position is (0, 0), that makes calculating the components much easier. A vector in standard position with a terminal point at (-3, 4) will have the components <-3, 4>. That means (A) is our answer.


  6. What are the components of a vector with initial point (-7, 2) and a terminal point of (-7, 8)?

    Correct Answer:

    <0, 6>

    Answer Explanation:

    Remember, we always subtract the initial point from the terminal point. Final minus initial. So -7 – (-7) = 0, but 8 – 2 = 6. That's (D). If we subtracted the points in the wrong order, we'd have (B), which is reversed! Since vectors are all about direction, we have to be really careful about how we handle the order of those initial and terminal (or terminal and initial?) points.


  7. A vector in standard form has the components <1, 8>. What is its initial point?

    Correct Answer:

    (0, 0)

    Answer Explanation:

    If anything, this question teaches us the importance answering the question that's asked. A vector in standard form with components <1, 8> has initial point (0, 0) and—Stop right there. That's all we need. Initial point of (0, 0). Why? Because it's in standard form. The end.


  8. A vector has an initial point (4, 3) and components of <4, 3>. Is this vector in standard form?

    Correct Answer:

    No, because its initial point is not at (0, 0)

    Answer Explanation:

    Vectors in standard form must have initial points at (0, 0). The vector we have starts at (4, 3). Only if the components of a vector match its terminal point is it considered standard. Even then, that's only because its initial point will have to be at (0, 0). Since this vector doesn't satisfy those requirements, it's not standard.


  9. A vector has an initial point at (-2, 1) and components of <9, -1>. What will be its terminal point?

    Correct Answer:

    (7, 0)

    Answer Explanation:

    Since we're given an initial point and the components, the terminal point is the addition of the coordinates of each. Algebraically speaking, if terminal point initial point = components, then components + initial point = terminal point. That means we have -2 + 9 = 7, and 1 + (-1) = 0. Our answer is very clearly (C).


  10. Vector s, in standard form, has a terminal point of (-1, 3). If vector t is equivalent to s and has a terminal point at (-7, -1), what is the initial point of t?

    Correct Answer:

    (-6, -4)

    Answer Explanation:

    Since s is in standard form, we know that its components are identical to its terminal points, <-1, 3>. If vector t is equivalent to s, it must have the same components as s. If its terminal point is at (-7, -1), it must have already "traveled" those components. To find its initial point, we can subtract the coordinates of the terminal point by the components of the vector. That means its initial point is -7 – (-1) = -6 and -1 – 3 = -4, or (D).


Aligned Resources

    More standards from High School: Number and Quantity - Vector and Matrix Quantities