The resultant of vectors A and B has a magnitude of 20 units.A has a magnitude of 8 units, and the angle between A and B is 40o. Calculate the magnitude of B
To calculate the magnitude of vector B, we can use the concept of vector addition and the relationship between magnitude, angle, and components of a vector.
1. Represent vector A by an arrow with a length of 8 units.
2. Draw vector B such that its tail is placed at the head of vector A, forming a triangle.
3. The magnitude of the resultant vector (A + B) is given as 20 units.
4. Using the given angle between vector A and B (40°), we can split vector B into two components: one parallel to vector A and one perpendicular to vector A.
5. The component of vector B parallel to vector A can be found using the formula B_parallel = B * cos(angle).
B_parallel = B * cos(40°)
6. The component of vector B perpendicular to vector A can be found using the formula B_perpendicular = B * sin(angle).
B_perpendicular = B * sin(40°)
7. Since the resultant vector (A + B) has a magnitude of 20 units, we can use the Pythagorean theorem to relate the components of vector B:
(B_parallel)^2 + (B_perpendicular)^2 = (20)^2
Substitute the values from steps 5 and 6:
(B * cos(40°))^2 + (B * sin(40°))^2 = (20)^2
8. Simplify the equation by expanding and combining terms:
(B^2 * cos^2(40°)) + (B^2 * sin^2(40°)) = 400
B^2 * (cos^2(40°) + sin^2(40°)) = 400
9. Apply the trigonometric identity: cos^2(theta) + sin^2(theta) = 1.
B^2 * 1 = 400
B^2 = 400
10. Take the square root of both sides to find the magnitude of vector B:
B = √400
B = 20
Therefore, the magnitude of vector B is 20 units.
law of sines
20/sin40=8/sinA
solve for angle A, then, you know angle B because the sume of the angles is 180
20/sin40=b/SinB