The question asks about two displacement vectors, A and B, and their resultant vectors C and D. Here's the breakdown:

• Given Vectors:

  • A = (3𝑖 - 4𝑗 + 4𝑘) m
  • B = (2𝑖 + 3𝑗 - 7𝑘) m

• Tasks:

  • (a) Calculate C = A + B and its magnitude.
  • (b) Calculate D = 2A - B and its magnitude.

(a) C = A + B

1. Add the components of A and B: $\mathbf{C} = (3 + 2)\mathbf{i} + (-4 + 3)\mathbf{j} + (4 - 7)\mathbf{k}$ $\mathbf{C} = 5\mathbf{i} - \mathbf{j} - 3\mathbf{k}$

2. Magnitude of C: $|\mathbf{C}| = \sqrt{(5)^2 + (-1)^2 + (-3)^2}$ $|\mathbf{C}| = \sqrt{25 + 1 + 9} = \sqrt{35} \approx 5.92 \, \text{m}.$

(b) D = 2A - B

1. First calculate 2A: $2\mathbf{A} = 2(3\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}) = 6\mathbf{i} - 8\mathbf{j} + 8\mathbf{k}.$

2. Subtract B: $\mathbf{D} = (6 - 2)\mathbf{i} + (-8 - 3)\mathbf{j} + (8 - (-7))\mathbf{k}$ $\mathbf{D} = 4\mathbf{i} - 11\mathbf{j} + 15\mathbf{k}.$

3. Magnitude of D: $|\mathbf{D}| = \sqrt{(4)^2 + (-11)^2 + (15)^2}$ $|\mathbf{D}| = \sqrt{16 + 121 + 225} = \sqrt{362} \approx 19.03 \, \text{m}.$

Final Results:

• C = $5\mathbf{i} - \mathbf{j} - 3\mathbf{k}$, $|\mathbf{C}| \approx 5.92 \, \text{m}$
• D = $4\mathbf{i} - 11\mathbf{j} + 15\mathbf{k}$, $|\mathbf{D}| \approx 19.03 \, \text{m}$

Would you like me to break down any specific step further?

5 Related Questions:

1. What is the geometric interpretation of vector addition and subtraction?
2. How do you compute the unit vector of a given vector?
3. What is the dot product of two vectors, and how does it differ from the cross product?
4. How do displacement vectors relate to real-world physics applications, like navigation?
5. Can the magnitude of a vector ever be negative?

Tip:

Always double-check the signs of vector components when performing addition or subtraction—they are a common source of mistakes.