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 is asking for the vector addition of four vector quantities A, B, C, and D. To solve this, we will add the corresponding components (i, j, and k) of each vector together.
Answer
$$ R = (R_x, R_y, R_z) $$ where $$ R_x = A_x + B_x + C_x + D_x, R_y = A_y + B_y + C_y + D_y, R_z = A_z + B_z + C_z + D_z $$
Answer for screen readers
The final vector after summing the components of vectors A, B, C, and D will be:
$$ R = (R_x, R_y, R_z) $$
Where:
- $R_x = A_x + B_x + C_x + D_x$
- $R_y = A_y + B_y + C_y + D_y$
- $R_z = A_z + B_z + C_z + D_z$
Steps to Solve
- Identify the Components of Each Vector
First, let's denote each vector's components.
Assume the vectors are defined as follows:
- Vector A: $A = (A_x, A_y, A_z)$
- Vector B: $B = (B_x, B_y, B_z)$
- Vector C: $C = (C_x, C_y, C_z)$
- Vector D: $D = (D_x, D_y, D_z)$
- Add the Components of the Vectors
Now, we will sum the components for each respective direction (i, j, and k).
The resulting vector $R$ will be calculated as:
$$ R = (A_x + B_x + C_x + D_x, A_y + B_y + C_y + D_y, A_z + B_z + C_z + D_z) $$
- Combine the Results into a Final Vector
After adding the respective components, we can combine the results to form the final vector:
$$ R = (R_x, R_y, R_z) $$
Where:
- $R_x = A_x + B_x + C_x + D_x$
- $R_y = A_y + B_y + C_y + D_y$
- $R_z = A_z + B_z + C_z + D_z$
The final vector after summing the components of vectors A, B, C, and D will be:
$$ R = (R_x, R_y, R_z) $$
Where:
- $R_x = A_x + B_x + C_x + D_x$
- $R_y = A_y + B_y + C_y + D_y$
- $R_z = A_z + B_z + C_z + D_z$
More Information
Vector addition is a fundamental operation in physics and engineering, as it allows for the combination of forces, velocities, and other vector quantities to find a resultant. Understanding how to break down vectors into their components is crucial for accurately solving problems involving multiple directions.
Tips
- Forgetting to add the components in each direction separately, which can lead to incorrect results.
- Incorrectly assuming that vectors can be added without considering their direction, particularly when they involve angles.
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