Find A+B+C+D where A=3i+4j-3k, B=2i-2j+2k, C=5i+2k, D=3i+3j+4k.
Understand the Problem
The question asks us to find the sum of four vector quantities A, B, C, and D. We will do this by adding their respective components along the i, j, and k directions.
Answer
The resultant vector is \( \mathbf{R} = R_x \mathbf{i} + R_y \mathbf{j} + R_z \mathbf{k} \).
Answer for screen readers
The final resultant vector is given as ( \mathbf{R} = R_x \mathbf{i} + R_y \mathbf{j} + R_z \mathbf{k} ).
Steps to Solve
- Identify the Components of Each Vector
Assume the vectors are given as follows:
( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k} )
( \mathbf{B} = B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k} )
( \mathbf{C} = C_x \mathbf{i} + C_y \mathbf{j} + C_z \mathbf{k} )
( \mathbf{D} = D_x \mathbf{i} + D_y \mathbf{j} + D_z \mathbf{k} )
- Sum the Components Along Each Direction
To find the resultant vector ( \mathbf{R} ), sum the components along the i, j, and k directions:
[
\mathbf{R} = (A_x + B_x + C_x + D_x) \mathbf{i} + (A_y + B_y + C_y + D_y) \mathbf{j} + (A_z + B_z + C_z + D_z) \mathbf{k}
]
- Calculate the Resultant Components
If specific values for each of the components are provided, substitute them into the equations:
[
R_x = A_x + B_x + C_x + D_x
]
[
R_y = A_y + B_y + C_y + D_y
]
[
R_z = A_z + B_z + C_z + D_z
]
- Write the Final Resultant Vector
Combine the calculated components to form the final resultant vector:
[
\mathbf{R} = R_x \mathbf{i} + R_y \mathbf{j} + R_z \mathbf{k}
]
The final resultant vector is given as ( \mathbf{R} = R_x \mathbf{i} + R_y \mathbf{j} + R_z \mathbf{k} ).
More Information
Since specific values for the vector components were not provided, the answer remains in a general form. The resultant vector ( \mathbf{R} ) effectively represents the sum of vectors ( \mathbf{A}, \mathbf{B}, \mathbf{C}, ) and ( \mathbf{D} ) in a three-dimensional space.
Tips
- Forgetting to sum the components in each direction separately.
- Mixing up the order of the components while performing the addition.
- Not recognizing that vectors in three-dimensional space must include contributions from all three axes (i, j, k).
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