A vector of magnitude 30 and direction eastwards is added with another vector of magnitude 40 and direction northwards. Find the magnitude and direction of resultant with the east.
draw a diagram, and you will see a 3-4-5 right triangle. So, the final displacement is 5, the hypotenuse.
The direction (measured from due east) is x, where tan(x) = 40/30
To find the magnitude and direction of the resultant vector, we can use vector addition.
Step 1: Convert the given magnitudes and directions to their respective components.
- The first vector has a magnitude of 30 and direction eastward. This means it has a horizontal component (in the east direction) of 30 and a vertical component (in the north direction) of 0.
- The second vector has a magnitude of 40 and direction northward. This means it has a horizontal component (in the east direction) of 0 and a vertical component (in the north direction) of 40.
Step 2: Add the corresponding components.
- Horizontal component: 30 + 0 = 30
- Vertical component: 0 + 40 = 40
Step 3: Use the Pythagorean theorem to find the magnitude of the resultant vector.
- The magnitude of the resultant vector is calculated as the square root of the sum of the squares of its components.
- Magnitude = sqrt((30^2) + (40^2))
- Magnitude ≈ sqrt(900 + 1600) ≈ sqrt(2500) ≈ 50
Step 4: Use trigonometry to find the direction of the resultant vector.
- The direction of the resultant vector is given by the angle it makes with the positive x-axis (eastward direction).
- To find this angle, we use the inverse tangent function: tan^-1(opposite/adjacent).
- direction = tan^-1(40/30)
- direction ≈ tan^-1(4/3) ≈ 53.13°
Therefore, the magnitude of the resultant vector is approximately 50 and the direction with respect to eastward is approximately 53.13°.