Vector Question!! (Physics)?

Question

1) A vector 5 cm long at 45 degrees. Find horizontal and vertical components graphically. Express as vectors (+/- for direction and X or Y with length and units).
2) What is resultant of vector 30 m/s East and 40 m/s south? (remember angle and magnitude)
How could Pythagorean theorem apply to number 2?

How do I do this?

Answer 1

Pythagorean Theorem:

[(a squared) + (b squared)] = c squared.

Question 1:

A vector of 5 cm long, we will call Vector c.

5 cm squared = 25. Divided by 2 is 12.5. The square root of 12.5 = 3.535

Proof:

(3.535 squared) + (3.535 squared) = 25

Graphically in matrix notation, the x sector and direction (+ or -) is listed first, then the y sector, i.e. (x,y)

So the two vectors to produce a 45 degree vector of 5 cm (northeast) would be expressed in matrix form as:

Vector a: (+3.535, 0) East
Vector b: (+3.535, +3.535) North

Question 2:

In matrix notation:

Vector a = (30, 0)
Vector b = (30, -40)

First, to find magnitude of Vector c, use Pythagorean formula

The square root of [(30 squared) + (40 squared)] = 50. That's your magnitude: 50 m/s

Now to find the angle:

All 3 angles of a triangle together must add to180 degrees. You begin with East and South vectors which we assume to be at 90 degrees to each other. So the other two angles must sum to 90 degrees. What are those angles?

Use the Sine Rule to find angles. (Small case letters = magnitude or side length; Upper case letters = Angle)

a / Sine A = b / Sine B = c / Sine C

a = 30 m/s East
b = 40 m/s South
c = 50 m/s Southeast
C = 90 degrees

Find B

b/Sine B = c/Sine C
40/Sine B = 50/Sine 90

Rule of inverse:
Sine B/40 = Sine 90 / 50

Multiplicative rule:
Sine B = [40 * Sine 90] / 50
Sine B = [40 * 1] / 50
Sine B = .80

Find Inverse Sine of .80
B = sin-1(.80)
B = 53.13 degrees (using scientific calculator)

A = 180 - B - C
A = 180 - 90 - 53.13
A = 36.70 degrees (for extra credit)

Answer: Resultant Southeast vector of 50 m/s at -53.13 degrees (below the x axis)

Answer 2

think of of it this sort: If I say i opt to run at 5 meters in keeping with 2nd, I even have given a importance (that's the size, for this reason extremely, the fee). that's unquestionably a scalar. yet you do no longer know which direction i bypass in till I say some thing like: i opt to run 5 meters in keeping with 2nd to the north. here, I particularly provide the importance AND the direction. that's unquestionably a vector. one million. stress is a vector because of the fact whenever you follow a stress to a minimum of a few thing, you're able to provide it a direction. (If I push down, then the direction is down) 2. undergo in strategies, there's a difference between speed and speed. speed is a vector. 3. Temperature isn't a vector because of the fact it would not have a direction. 4. Acceleration is a vector. here is an occasion for you: Mass vs weight. Mass is a scalar, weight is a vector! once you're saying this merchandise is 5kg, it relatively is the comparable no rely what direction you provide it. Weight is diverse because of the fact it refers back to the stress that gravity imposes on an merchandise. considering that gravity is a stress, that's a vector quantity.

Answer 3

1) x= 5 /2*sqrt(2)
y=5/2*sqrt(2)

30m/s and 40 are legs of a right angle triangle
resultant in magnitude = sqrt( 30^2+40^2) =50m/s

tan @= 4/3 s0 @= -53.13º with the positive direction (east)
supposing North upwards