(a) Find the magnitude and direction of the vector R⃗ that is the sum of the three vectors A⃗, B

(a) Find the magnitude and direction of the vector R⃗ that...

Key Concepts

Graphical Representation of Vectors

Graphical methods such as the tip-to-tail method are essential for visualizing vector addition and subtraction. They provide a clear illustration of how vectors combine, showing the consecutive placement of vectors and the ultimate determination of the resultant vector's magnitude and direction.

Vector Addition

Vector addition involves combining two or more vectors to yield a single resultant vector. This process is done graphically by placing the tail of one vector at the tip of another (the tip-to-tail method), and algebraically by summing their components. The resultant vector summarizes the net effect of the individual vectors applied together.

Vector Subtraction

Vector subtraction is the process of determining the difference between two vectors. It can be viewed as adding the negative of a vector to another vector, effectively reversing the direction of one vector before addition. This operation is useful for finding relative displacement or change between vectors.

Resultant Vector

The resultant vector is the single vector that has the same effect as the combined influence of several individual vectors. It is characterized by its magnitude and direction, both of which can be calculated either graphically or using algebraic methods such as components or trigonometry.

Transcript

00:01 Our question wants us to look at figure e1 .28 and get a resulting vector from adding vector a, b, and c together, as well as getting a resulting vector from subtracting c minus a minus b.

00:16 Okay.

00:16 And so vector a, b, and c, i wrote out their magnitudes here, and their angles i wrote relative to the positive x axis.

00:23 So vector a, for instance, is along the negative y axis.

00:27 So relative to the positive x axis, that's negative 90 degrees.

00:31 Or positive 270 degrees.

00:33 They give you the same answer either way.

00:35 Theta b is 30 degrees to the right or 30 degrees off of the y -axis.

00:41 So that would be 90 minus 30 or 60 degrees from the positive y -axis.

00:45 And the angle c is 25 degrees below the negative x -axis, so that you have to add 180 degrees to that to get the difference from the positive x -axis.

00:55 So that would be 205 degrees.

00:57 So then for part a, it wants us to find the magnitude and direction of vector r when vector r is defined as vector a plus vector b plus vector c.

01:09 And then using that, it wants us to also draw out a rough sketch of a diagram of vector a plus vector b plus vector c to show that it does indeed add up to the values we got for vector r.

01:20 So when you're adding vectors together, you can only add light components.

01:25 So what we end up with is we have, of course, vector r is made up of its x component and its y components.

01:33 So this is going to be r of x plus r of y, but r of x is going to be all the x components of a, b, and c added together.

01:40 So this is going to be ax plus bx plus cx, and then r of y is going to be all the like components of a, b, or all the y components of a, b, and c added together.

01:58 So we ran out of room here, but we'll just write it below.

02:02 So this is going to be ay plus by plus cy.

02:11 And of course, everything in this parenthesis would go right here.

02:17 Okay.

02:18 So then the question is, what's ax, b, x, cx? we can use those to find r sub x.

02:23 And what's a, y, b, y, c, y, we can use those to find r sub y, and then use r sub y and r sub x to find the magnitude of r as well as the direction of the vector r.

02:32 Okay.

02:33 So we ran out of room a little bit.

02:34 There, but we can do it over here.

02:35 So now we end up with r is equal to.

02:39 Well, let's start with the x components.

02:41 So it's going to be a times the cosine of theta a.

02:45 And instead of writing out cosine, i'll just write c for my abbreviation for cosine.

02:51 So it's going to be a times the cosine of theta a plus b times the cosine of theta b, plus c times the cosine of theta b.

03:10 Okay, so that would be all of our x components, so that's in the i -hat direction.

03:17 Plus, now we have to add in all of our y components.

03:21 So this is going to be b, excuse me, we'll start with a again.

03:25 So this is going to be a times the sign, which i'll abbreviate with just s, times the sine of theta a plus b times the sign of theta b, plus c times the sign of theta c.

03:51 We plug all these values into this expression.

03:53 Of course, this is going to be the jihad direction.

03:56 We find that all of the x components add up to be negative 3 .37, and the units here are meters in the i -hat direction, minus 0 .08 meters in the j -hat direction.

04:18 So now we can find the magnitude of r using pythagorean theorem.

04:23 It's the square root of r subx squared plus r sub y squared, where r sub x is negative 3 .37, and r sub y is negative 0 .08.

04:34 So plugging those values in, we find that the magnitude of r is equal to 11 .4 meters.

04:51 Now using our trigonometric identities, the tangent of the angle theta r is going to be equal to the opposite over the adjacent.

04:58 So that's going to be r sub y over r sub x.

05:01 Or in other words, theta r is equal to inverse tangent of r sub y over r sub x.

05:18 So plugging those values into this expression, we find that this is equal to 1 .36 degrees.

05:25 But we have to consider the fact that r sub x and r sub y are both negative, right? so if they're both negative, that means they're both in quadrant three.

05:36 So 1 .36 degrees is in quadrant 1 .3 .1 .1 .1 .1 .1 .1.

05:39 So what we found was actually the angle.

05:41 Of the right triangle that makes up r relative to r sub x and r sub y.

05:45 So we need to actually add 180 degrees to this to get us from quadrant 1 to quadrant 3, where we are supposed to be since r sub y and r sub x are both negative.

05:56 So what we end up with is an actual value for theta r relative to the positive x axis of 181 .4 degrees, which we can box in as the next part of our solution for part a.

06:10 And lastly, for part a, it wants us to draw a diagram to show that these vectors do indeed add up to the resulting vector that we found...

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