Vector Sheet: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Go up
- by Chris Murray, But Mostly Brad Hamlin 2002

1. Convert this vector to Vector Component form: The first thing to do is draw the components: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

vx = (15 m/s)cos(29o) = 13.12 m/s

The y component (red) in this case is up, which is the positive direction for y, and is the side opposite to the known angle, so we use sine to find it:

vy = (15 m/s)sin(29o) = 7.27 m/s

And finally we can write it in VC notation:

v = 13.12 m/s x + 7.27 m/s y

2. Convert this vector to Vector Component form: The first thing to do is draw the components: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

sx = (300 m)cos(40o) = 230 m

The y component (red) in this case is up, which is the positive direction for y, and is the side opposite to the known angle, so we use sine to find it:

sy = (300 m)sin(40o) = 193 m

And finally we can write it in VC notation:

s = 230 m x + 193 m y

3. Convert this vector into vector component notation: The first thing to do is draw the components: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side opposite to the known angle, so we use sine to find it:

vx = (32.5 m/s)sin(55o) = 18.6 m/s

The y component (red) in this case is down, which is the negative direction for y, and is the side adjacent to the known angle, so we use cosine to find it:

vy = (32.5 m/s)cos(55o) = 26.6 m/s

And finally we can write it in VC notation:

v = 18.6 m/s x - 26.6 m/s y

4. Convert this vector into vector component notation: The first thing to do is draw the components: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the left, which is the negative direction for x, and is the side opposite to the known angle, so we use sine to find it:

vx = (120 m/s)sin(38o) = 73.8 m/s

The y component (red) in this case is down, which is the negative direction for y, and is the side adjacent to the known angle, so we use cosine to find it:

vy = (15 m/s)cos(38o) = 94.6 m/s

And finally we can write it in VC notation:

v = -73.8 m/s x - 94.6 m/s y

5. Convert this vector into vector component notation: The first thing to do is draw the components: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the left, which is the negative direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

sx = (132 m)cos(18o) = 126 m

The y component (red) in this case is up, which is the positive direction for y, and is the side opposite to the known angle, so we use sine to find it:

sy = (132 m)sin(18o) = 40.8 m

And finally we can write it in VC notation:

s = -126 m x + 40.8 m y

6. Convert this vector into vector component notation: The first thing to do is draw the components: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the left, which is the negative direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

ax = (160 m/s/s)cos(30o) = -140 m/s/s

The y component (red) in this case is down, which is the negative direction for y, and is the side opposite to the known angle, so we use sine to find it:

ay = (160 m/s/s)sin(30o) = 80 m/s

And finally we can write it in VC notation:

v = -140 m/s/s x - 80. m/s/s y

7. Convert 5.0 m x + 6.0 m y to Angle Magnitude notation:

The first thing to do is to draw the vector itself: Here, the blue is the x component, and the red is the y.  The black vector is the vector sum of the two or "the vector"

The magnitude of the vector is the hypotenuse of the triangle:

hyp =
Ö{(5.0 m)2 + (6.0 m)2} = 7.81 m = 7.8 m

The angle between the x axis (blue) and the vector (black) is the inverse tangent of opposite over adjacent:

q =Tan-1{6.0 m/5.0 m} = 50.2o

8. Convert -3.00m x+ 7.00m y into angle magnitude notation:

First, we draw the vector. Here, the blue is the x component, and the red is the y component.  The black is the actual vector that we are trying to find.

The magnitude of that vector is the hypotenuse of the triangle, so Ö{32+92}= 7.62m.

The angle between the x- axis and the vector is simply the inverse tangent of opposite over adjacent.

q =
Tan-1{7.00 m/3.00 m} = 66.8o

9. Convert -4.2m x - 3.2m y into angle magnitude notation

First, we draw the vector Here, blue is the x- component, and red is the y- component. The black is our actual vector.

The magnitude of the vector is the hypotenuse of the triangle, so
Ö{3.22+4.22}= 5.3m

The angle between the x- axis and the vector is simply the inverse tangent of opposite over adjacent.

q =Tan-1{3.2 m/4.2 m} = 37o

10. Convert 1.12m x - 5.70m y into angle magnitude notation

First, we draw the vector Here, blue is the x- component, and red is the y- component. The black is our actual vector.

The magnitude of the vector is the hypotenuse of the triangle, so
Ö{1.122+5.702}= 5.81m

The angle between the x- axis and the vector is simply the inverse tangent of opposite over adjacent.

q =Tan-1{5.70 m/1.12 m} = 78.9o

11. Add vectors #7 & #8 from part #2.

 5.0m x + 6.0m y + -3.00m x + 7.00m y 2.0m x + 13.0m y

12. Subtract Vector #8 from #7.

 5.0m x + 6.0m y +-(-3.00m) x + -(7.00m) y 8.0m x - 1.0m y

13. Subtract vector #8 from vector #9.

 -4.2m x - 3.2m y +-(-3.00m) x + -(7.00m) y -1.2m x - 10.2m y

14. Add vectors #10 & #9.

 1.12m x - 5.70m y -4.2m x - 3.2m y -3.1m x - 8.9m y

15. You have a velocity of #1 for 3.75 seconds.  What is your displacement in VC notation?

Well, you take what you have already done in #1, which is convert the vector into angle magnitude, and multiply those numbers by the time to find the distance.
In numbers,

 3.75s(13m/s) x + 3.75(7.3m/s) y 49m x + 27m y

16. You undergo the displacement as in #2 in 112 seconds.  What is your velocity in VC notation?

You take the vector components, and you divide them by 112.

 (230m)/112s x + (190m)/112s y 2.1m/s x + 1.7m/s y

17. You have an initial velocity of #4, and you accelerate like #6 for .126 seconds.  What is your final velocity?

First, we take down all the known data.

 x y S= N/A N/A U= -73.8m/s -94.6m/s V= ? ? A= -140m/s/s -80m/s/s T= .126s .126s

Then, we use V=U+AT twice, once in the x- direction and once in the y- direction.

X- DIRECTION

V=-73.8+-140(.126)
V=-91m/s

Y- DIRECTION

V=-94.6+-80(.126)
V=-105m/s

So, the final vector in VC notation would look like:

-91m/s x - 105m/s y

18. You undergo the displacement in #8 in 1.38 seconds.  What is your velocity in AM notation?

We just divide the AM version of #8 by 1.38 to get m/s.  So, 7.62m/1.38s= 5.52m/s @ 66.8o

19. Add these angle magnitude vectors analytically, and express their sum as an angle magnitude vector.

A: B: First, we convert them into vector component vectors, then add them, and then convert them back into angle magnitude vectors.
So, step one:

A: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

sx = (12 m)cos(25o) = 10.9 m

The y component (red) in this case is up, which is the positive direction for y, and is the side opposite to the known angle, so we use sine to find it:

sy = (12 m)sin(25o) = 5.1 m

And finally we can write it in VC notation:

A = 10.9 m x + 5.1 m y

B: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

sx = (10. m)cos(48o) = 6.69 m

The y component (red) in this case is down, which is the negative direction for y, and is the side opposite to the known angle, so we use sine to find it:

sy = (10.m)sin(48o) = 7.43 m

And finally we can write it in VC notation:

B = 6.7 m x - 7.4 m y

So now we can add A & B:

 10.9m x + 5.1m y +6.7m x - 7.4m y 17.6m x - 2.3m y

Then, we change that vector component vector back into an angle magnitude vector. Here, the blue is the x component, and the red is the y.  The black vector is the vector sum of the two or "the vector"

The magnitude of the vector is the hypotenuse of the triangle:

hyp =
Ö{(17.6 m)2 + (-2.3 m)2} =  17.74m = 18 m

The angle between the x axis (blue) and the vector (black) is the inverse tangent of opposite over adjacent:

q =Tan-1{2.3 m/17.6 m} = 7.7o

So, the answer is 18m, 7.7 degrees below the x-axis.

20. Add these two vectors analytically, and express their sum as an Angle-Magnitude vector.

A: B: First, we convert them into vector component vectors, then add them, and then convert them back into angle magnitude vectors.

A: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side adjacent to the known angle, so we use cosine to find it:

sx = (32 m)cos(30o) = 27.7 m

The y component (red) in this case is up, which is the positive direction for y, and is the side opposite to the known angle, so we use sine to find it:

sy = (32 m)sin(30o) = 16 m

And finally we can write it in VC notation:

A = 27.7 m x + 16 m y

B: The blue component is the x component, and the red is the y component.

The x component (blue) in this case is to the right, which is the positive direction for x, and is the side opposite to the known angle, so we use sine to find it:

sx = (40. m/s)sin(15o) = 10.35 m

The y component (red) in this case is down, which is the negative direction for y, and is the side adjacent to the known angle, so we use cosine to find it:

sy = (40. m/s)cos(15o) = 38.64 m

And finally we can write it in VC notation:

B = 10.4 m x - 38.6 m y

So now we can add A & B:

 27.7m x + 16m y +10.4m x - 38.6m y 38.1m x - 22.6m y

Then, we change that vector component vector back into an angle magnitude vector. Here, the blue is the x component, and the red is the y.  The black vector is the vector sum of the two or "the vector"

The magnitude of the vector is the hypotenuse of the triangle:

hyp =
Ö{(38.1 m)2 + (22.6 m)2} =  44.29m = 44 m

The angle between the x axis (blue) and the vector (black) is the inverse tangent of opposite over adjacent:

q =Tan-1{22.6m/38.1 m} = 30.67o

So, the answer is 44m, 31 degrees below the x-axis.