Vectors: the component method?

Question

Please explain fully, I just don't get the concept. Thanks all.

Answer 1

Vectors have magnitude and direction. So, let's look at what that means.

Assume a right triangle that looks like:
A
^
|
|
|<------- B

Assume arrow A is aimed straight up; so it's angle relative to the horizontal is theta = 90 deg. [By convention, we measure degrees in the counter-clockwise direction from the horizontal or X axis.] Assume the magnitude of A is a = 4 units.

Similarly, assume arrow B is aimed straight left; so it's angle relative to the horizontal is omega = 180 deg. And B's magnitude is b = 3 units.

Now, what's the sum A + B? Remember, both of these guys are vectors; so the sum of two vectors is a vector. Let's call the sum, vector C. So we have A + B = C But, wait, A and B are components of C. That is, C is made up of two parts we call vector A and vector B. And there you have it...components of a vector (e.g., C) are in and of themselves also vectors (e.g., A and B).

So, what about C? As a vector, it has both magnitude (c) and direction, call it angle psi. How do we get c and psi from A + B = C?

Well, a and b form the sides of a right triangle. We specified that. So the distance between A and B is found from c^2 = a^2 + b^2. If you work this, you'll find c = 5 units; that's the magnitude of C in this case.

This only works when we form a right triangle with the components; however, we can always specify any vector as the sum of its X and Y components...components that go horizontally (the X's) and vertically (the Y's). This means, we can always form right triangles with the components of a given vector. In other words, we can always find a vector's magnitude through c^2 = a^2 + b^2 where a and b are the magnitudes of vector C's vertical and horizontal components.

So we have the magnitude c = 5. But what's the direction of C? Again we invoke the fact that A and B form a right triangle. We also recall that psi, the direction angle is measured counter clockwise from the horizontal. So, just by looking at the picture, we see that psi > pi/2 which is 90 degrees. In fact psi = 180 - arcsin(4/5) = 180 - 53 ~ 127 degrees as measured CCW from the horizontal.

In sum, C = 5 units at 127 degrees when A = 4 at 90 deg and B = 3 at 180 degrees. From the graph, the C answer is intuitively pleasing because, as the hypoteneuse of a right triangle, clearly the magnitude of C is larger than any of its components (the sides of the triangle). 5 is bigger than 3 and 4. And the direction of C has to lie between the direction of B at 180 degrees and of A at 90 degrees. 127 degrees does just that.

Answer 2

Well set up a graph. Draw a line between (0, 0) and (1,1)

Call that line V, your vector

well the x-component would be the distance along the x axis
i.e. 1 unit

and the y-component would be a distance of 1 unit along the y-axis

Using Pythagoras

V² = (1)² + (1)²

v = sqrt (2)

So your components would be V = 1i + 1j (let x = i, y = j)

the magnitude of V = sqrt (2)