Question

5) If A= (51 i – 14 j + 43k ) cm and B = ( -31i -21j-11k) mm ;
a) Find D= -A+3B.
b) Find the magnitude D.

Answer 1

Answer on Question #58827 – Math – Analytical Geometry

Question

5) If $\vec{A} = (51\hat{i} - 14\hat{j} + 43\hat{k})\,cm$ and $\vec{B} = (-31\hat{i} - 21\hat{j} - 11\hat{k})\,mm$.

a) Find $\vec{D} = -\vec{A} + 3\vec{B}$.

b) Find the magnitude $\vec{D}$.

Solution

a) From the condition of the question we can see that the components of vector $\vec{A}$ are given in centimeters and the components of vector $\vec{B}$ are given in millimeters. Let's first convert them into meters:

$$\vec{A} = (0.51\hat{i} - 0.14\hat{j} + 0.43\hat{k})\,m, \vec{B} = (-0.031\hat{i} - 0.021\hat{j} - 0.011\hat{k})\,m.$$

To find $\vec{D} = -\vec{A} + 3\vec{B}$, we first multiply vectors $\vec{A}$ and $\vec{B}$ by scalars $-1$ and $3$ respectively. Then, we add these two vectors.

Let's multiply vectors $\vec{A}$ and $\vec{B}$ by scalars $-1$ and $3$ respectively:

$$-1 \cdot \vec{A} = (-0.51\hat{i} + 0.14\hat{j} - 0.43\hat{k})\,m,$$ $$3 \cdot \vec{B} = (-0.093\hat{i} - 0.063\hat{j} - 0.033\hat{k})\,m.$$

Using formulas

$$\vec{A} = (A_x \hat{i} + A_y \hat{j} + A_z \hat{k}),$$ $$\vec{B} = (B_x \hat{i} + B_y \hat{j} + B_z \hat{k}),$$ $$\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} + (A_z + B_z) \hat{k}$$

(when we add two vectors, we must add the components separately).

Let's add vectors $-1 \cdot \vec{A}$ and $3 \cdot \vec{B}$:

$$\vec{D} = -\vec{A} + 3\vec{B} = (-0.51 + (-0.093))\hat{i} + (0.14 + (-0.063))\hat{j} + \\ + (-0.43 + (-0.033))\hat{k} = (-0.603\hat{i} + 0.077\hat{j} - 0.463\hat{k})\,m.$$ $$\vec {D} = - \vec {A} + 3 \vec {B} = (- 0.603 \hat {i} + 0.077 \hat {j} - 0.463 \hat {k}) \, m.$$

b) The magnitude of the vector $\vec{D}$ can be found using the Pythagorean theorem:

$$\left| \vec {D} \right| = \sqrt {D _ {x} ^ {2} + D _ {y} ^ {2} + D _ {z} ^ {2}} = \sqrt {(- 0.603 \, m) ^ {2} + (0.077 \, m) ^ {2} + (- 0.463 \, m) ^ {2}} = 0.764 \, m.$$

Answer:

a) $\vec{D} = -\vec{A} + 3\vec{B} = (-0.603\hat{i} + 0.077\hat{j} - 0.463\hat{k}) \, m.$

b) $\left| \vec{D} \right| = 0.764 \, m.$

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