Three vectors of lengths A = 45.6, B = 24.2, and C = 30.4. The angles are θa = 29.6° and θb = 57.5°, and C points along the negative y-axis.
Determine the length of the vector A - C.
Calculate the angle of this vector.
Determine the length of the vector A - B + C.
Calculate the angle of this vector.
Determine the length of the vector A + B - C.
Calculate the angle of this vector.
Determine the length of the vector B - 2A.
Calculate the angle of this vector.
To answer these questions, we can use vector addition and subtraction along with trigonometric functions. Let's go step by step:
1. To determine the length of vector A - C, we subtract the components of vector C from vector A. Since C points along the negative y-axis, its components are (0, -C). The components of A are (Ax, Ay), where Ax is the horizontal component and Ay is the vertical component. So, the components of A - C are (Ax, Ay) - (0, -C) = (Ax, Ay + C). The length of the vector A - C can be calculated using the formula: sqrt(Ax^2 + (Ay + C)^2).
2. To calculate the angle of the vector A - C, we can use the inverse tangent function. The angle can be determined as: atan((Ay + C)/Ax).
3. To determine the length of vector A - B + C, we first need to find the components of A - B. Subtract the components of vector B from vector A. The components of A - B are (Ax - Bx, Ay - By). Then, add the components of vector C to it. The components of A - B + C are (Ax - Bx, Ay - By) + (0, -C) = (Ax - Bx, Ay - By - C). The length of the vector A - B + C can be calculated using the formula: sqrt((Ax - Bx)^2 + (Ay - By - C)^2).
4. To calculate the angle of the vector A - B + C, we can use the inverse tangent function. The angle can be determined as: atan((Ay - By - C)/(Ax - Bx)).
5. To determine the length of vector A + B - C, we first need to find the components of A + B. Add the components of vector B to vector A. The components of A + B are (Ax + Bx, Ay + By). Then, subtract the components of vector C from it. The components of A + B - C are (Ax + Bx, Ay + By) - (0, -C) = (Ax + Bx, Ay + By + C). The length of the vector A + B - C can be calculated using the formula: sqrt((Ax + Bx)^2 + (Ay + By + C)^2).
6. To calculate the angle of the vector A + B - C, we can use the inverse tangent function. The angle can be determined as: atan((Ay + By + C)/(Ax + Bx)).
7. To determine the length of vector B - 2A, we need to multiply the components of vector A by -2 and then subtract them from the components of vector B. The components of B - 2A are (Bx - 2Ax, By - 2Ay). The length of the vector B - 2A can be calculated using the formula: sqrt((Bx - 2Ax)^2 + (By - 2Ay)^2).
8. To calculate the angle of the vector B - 2A, we can use the inverse tangent function. The angle can be determined as: atan((By - 2Ay)/(Bx - 2Ax)).
By plugging in the values of the given vectors and angles, you can compute the lengths and angles using the above formulas.