vector A = a displacement of 20 m due east
vector B = a displacement of 30 m north
find the magnitude and direction of the vector sum a+b
To find the magnitude and direction of the vector sum of A and B (A + B), we can use the Pythagorean theorem and trigonometry.
Step 1: Determine the vector components of A and B.
- Vector A has a displacement of 20 m due east, which corresponds to a displacement of 20 m in the x-direction.
- Vector B has a displacement of 30 m north, which corresponds to a displacement of 30 m in the y-direction.
So, the vector components of A and B are:
- A = 20i (i component represents the x-direction)
- B = 30j (j component represents the y-direction)
Step 2: Calculate the vector sum of A + B.
Add the vector components of A and B to get the vector sum:
A + B = (20i) + (30j) = 20i + 30j
Step 3: Calculate the magnitude of the vector sum.
The magnitude of a vector can be found using the Pythagorean theorem:
|A + B| = √((20^2) + (30^2))
Calculating this, we have:
|A + B| = √(400 + 900)
|A + B| = √1300
|A + B| = 36.06 m (rounded to two decimal places)
Step 4: Calculate the direction of the vector sum.
The direction of the vector sum can be found using trigonometry. We can use the inverse tangent function:
θ = tan^(-1)((Δy)/(Δx))
Calculating this, we have:
θ = tan^(-1)((30)/(20))
θ ≈ 56.31° (rounded to two decimal places)
The magnitude of the vector sum is 36.06 m and the direction is approximately 56.31° (counterclockwise from the positive x-axis).
To find the magnitude and direction of the vector sum, A + B, we can use the Pythagorean theorem and trigonometry.
First, let's draw a diagram to visualize the situation. Place vector A horizontally (representing east) and vector B vertically (representing north). The two vectors will form a right-angled triangle.
To find the magnitude of the vector sum, we will use the Pythagorean theorem:
Magnitude = √(A^2 + B^2)
In this case, A = 20 m (east) and B = 30 m (north). So,
Magnitude = √((20)^2 + (30)^2)
= √(400 + 900)
= √1300
≈ 36.06 m
Next, to find the direction of the vector sum, we will use trigonometry. We can use the tangent function to calculate the angle:
tan(θ) = opposite/adjacent
In this case, the opposite side is B (30 m north) and the adjacent side is A (20 m east). Therefore,
tan(θ) = B/A
= 30/20
= 1.5
To find the angle θ, take the inverse tangent (arctan or tan^-1) of 1.5:
θ = tan^-1(1.5)
≈ 56.31°
So, the magnitude of the vector sum A + B is approximately 36.06 m, and the direction is approximately 56.31° north of east.
sqrt (30^2+20^2) =sqrt(1300) = 10 sqrt(13)
tan A = 2/3
where A is compass angle CLOCKWISE from north