Question

if \\(\vec{d} = (5.00\hat{i} - 3.00\hat{j})\\,\text{m}\\), \\(\vec{b} = (-4.00\hat{i} + 6.00\hat{j})\\,\text{m}\\), and \\(\vec{a} = (4.00\hat{i} - 24.0\hat{j})\\,\text{m}\\), find the unknown constants \\(a\\) and \\(b\\) such that \\(a\vec{d} + b\vec{b} + \vec{a} = \vec{0}\\).
\\(a = \square\\)
\\(b = \square\\)

Explanation:

Step1: Substitute vectors into equation

$$a(5.00\hat{\mathbf{i}} - 3.00\hat{\mathbf{j}}) + b(-4.00\hat{\mathbf{i}} + 6.00\hat{\mathbf{j}}) + (4.00\hat{\mathbf{i}} - 24.0\hat{\mathbf{j}}) = 0$$

Step2: Group $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ components

For $\hat{\mathbf{i}}$:

$$5.00a - 4.00b + 4.00 = 0$$

For $\hat{\mathbf{j}}$:

$$-3.00a + 6.00b - 24.0 = 0$$

Step3: Simplify the $\hat{\mathbf{j}}$ equation

Divide by $-3.00$:
$$a - 2.00b + 8.00 = 0 \implies a = 2.00b - 8.00$$

Step4: Substitute $a$ into $\hat{\mathbf{i}}$ equation

$$5.00(2.00b - 8.00) - 4.00b + 4.00 = 0$$
$$10.0b - 40.0 - 4.00b + 4.00 = 0$$
$$6.00b - 36.0 = 0$$

Step5: Solve for $b$

$$6.00b = 36.0 \implies b = \frac{36.0}{6.00} = 6.00$$

Step6: Solve for $a$ using $a=2.00b-8.00$

$$a = 2.00(6.00) - 8.00 = 12.0 - 8.00 = 4.00$$

Answer:

$a = 4.00$, $b = 6.00$