Vectors in the Coordinate Plane: Vectors in the Coordinate Plane
Concept #1: Component Form of a Vector: Concept #1: Component Form of a Vector “Component form” means to describe a vector using its terminal point. We get component form by finding the difference in the coordinates of the initial point and the terminal point. It’s like finding slope, but written differently. If we consider the initial point to have the coordinates (x 1 , y 1 ) and the terminal point to have the coordinates (x 2 , y 2 ), then the component form of the vector would be Note: these are NOT coordinates! That’s why we have pointy thingamajigs and not parentheses. This refers to the difference between the x and y coordinates. The terminal point is ALWAYS used first.
Slide 3: Think about this graphically. Let’s say we have a vector with an initial point at (0, 0) and a terminal point at (3, 6). Its terminal point is represented by the distance between the x-values, and the distance between the y-values. In this case, that happens to be the coordinates of the terminal point. However, what happens if the vector does not start at the origin?
Slide 4: This vector has an initial point at (3, -3) and a terminal point at (7, -1). The component form of this vector would NOT be (7, -1). Component form describes the distance between the initial point and the terminal point, using the x and y values of each.
So, to recap…: So, to recap… Component form is not usually going to be the coordinates of the terminal point. Each vector is uniquely described by its component form, but the component form does not mean a unique vector. When doing your calculations, ALWAYS set the terminal point as the first numbers. Terminal coordinates
Examples: Examples Find the component forms of the vectors with the given initial and terminal points. (A is initial, B is terminal.) A(2, -7), B(-6, 9) A(-4.3, 1.8), B(9.4, -6.2) A(-5, -9), B(-11, -10)
Concept #2: Magnitude in the Coordinate Plane: Concept #2: Magnitude in the Coordinate Plane Magnitude is simply the distance from the initial point to the terminal point. We find it using the Distance Formula. We can either find it using the coordinates of the initial point and the terminal point, or by using the component form. Does it matter which way you choose? Not really.
To find the magnitude:: To find the magnitude: Using initial and terminal points: Remember, this is NOT absolute value. They’re both positive, because they’re distances, but this is read “magnitude of vector v”. Using component form
Examples: Examples Find the magnitude of A(5, -8), B(14, -3) A(1, 11), B(-5, 6)
Concept #3: Vector Operations: Concept #3: Vector Operations We are able to add and subtract vectors from one another. This would give us a resultant vector, just like we got using the triangle method or parallelogram method. We are also able to multiply a vector by a scalar. When working with vectors, -b means a vector with the same magnitude as b but going in the exact opposite direction. You would be, in effect, adding a vector, not subtracting it. 3b, for example, would be a vector whose magnitude is three times the magnitude of b . If you were asked to draw a graphical representation, all you’d have to do is draw b , then draw whatever happened to it..
Vector Operation Rules: Vector Operation Rules
Slide 12: Think about this graphically. If we had vector a , with initial point at (0,0) and terminal point at (4, 3), and vector b, with initial point at (0,0) and terminal point at (-4, 3), we could use the tail-to-tail method (parallelogram method) to find the resultant vector. The resultant vector has a terminal point at (0, 6). That could have been obtained by adding the component forms of vectors a and b .
Examples: Examples Find each of the following for x + y 2y – 3z -4x – 5z + y
Concept #4: Unit Vectors: Concept #4: Unit Vectors A unit vector, kind of like the unit circle, has a magnitude of 1. We sometimes have a need to work with a unit vector that is going the same direction as another vector. In technical terms, the unit vector is a scalar multiple of a vector. In non-technical terms, divide the component form of vector v by the magnitude of vector v .
The Official Formula: The Official Formula To finish the problem, just remember that dividing by the magnitude is the same thing as multiplying by a scalar, and divide each value of the component form by the magnitude.
Example 1: Example 1 Find a unit vector u with the same direction as v =
More Examples: More Examples Find a unit vector u with the same direction as v = Find a unit vector u with the same direction as v =
More on unit vectors: More on unit vectors The unit vector that goes on the positive x-axis is referred to as i , and the unit vector that goes on the positive y-axis is referred to as j. These are known as the standard unit vectors . By using scalar multiplication, we can express any vector as th e sum of these two vectors. When we do this, the vector sum is known as a linear combination, and is written as ai + bj .
Slide 19: As we showed earlier in this presentation, the resultant vector (the red one) is simply the sum of the component forms of the other two vectors. What if the red vector was longer? The unit vectors can not be any longer, by definition, but if we imagined them to be multiplied by a scalar, then they would be longer. Because of this, if we simply multiply a scalar by the unit vectors, we get the longer resultant vector.
Example 1: Example 1 Let be the vector with the given initial and terminal points. Write as a linear combination of the unit vectors. D(3, -2), E(1, 4) First, find the component form. Next, write as a linear combination. -2i + 6j
Examples: Examples Let be the vector with the given initial and terminal points. Write as a linear combination of the unit vectors. D(-6, 0), E(2, 5) D(-3, -8), E(-7, 1)
Concept #6: Finding Component Form given a magnitude and direction angle: Concept #6: Finding Component Form given a magnitude and direction angle This is based off of finding the rectangular components of a vector. To find x and y, we used trigonometry. Every single time, you just multiply the hypotenuse by the cosine of the angle to find x, and by the sine of the angle to find y. That’s all we have to do here, too.
Slide 23: The hypotenuse is simply the magnitude of the vector. If we want to find the component form, that’s the same thing as finding the change in x—the length of the x side—and the change in y—the length of the y side, and putting those together. All we have to do is multiply the magnitude by the cosine and sine of the angle.
Example 1: Example 1 Find the component form of the vector with magnitude 16 and direction angle 110 °.
Concept #7: Finding Direction Angle: Concept #7: Finding Direction Angle Using the same logic as the last concept, if we broke down the vector into a right triangle, to find the missing angle, all we’d have to do is use some simple trigonometry.