If `vec(A) = 5 hat(i) - 3 hat(j) + 4 hat(k) and vec(B) = hat(j) - hat(k) ` , find the sine of the angle between `vec(a) and vec(B) `

Explanation

To find the sine of the angle between the vectors \(\vec{A}\) and \(\vec{B}\), we will use the formula involving the cross product of the two vectors. The formula states:

\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]

From this, we can express \(\sin \theta\) as:

\[ \sin \theta = \frac{|\vec{A} \times \vec{B}|}{|\vec{A}| |\vec{B}|} \]

### Step 1: Define the vectors
Given:
\[ \vec{A} = 5 \hat{i} - 3 \hat{j} + 4 \hat{k} \]
\[ \vec{B} = \hat{j} - \hat{k} \]

### Step 2: Calculate the cross product \(\vec{A} \times \vec{B}\)
Using the determinant method to find the cross product:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 5 & -3 & 4 \\ 0 & 1 & -1 \end{vmatrix} \]

Calculating the determinant:

1. For \(\hat{i}\):
\[ \hat{i} \left((-3)(-1) - (4)(1)\right) = \hat{i}(3 - 4) = -\hat{i} \]

2. For \(\hat{j}\):
\[ -\hat{j} \left((5)(-1) - (4)(0)\right) = -\hat{j}(-5) = 5\hat{j} \]

3. For \(\hat{k}\):
\[ \hat{k} \left((5)(1) - (-3)(0)\right) = \hat{k}(5) = 5\hat{k} \]

Putting it all together:
\[ \vec{A} \times \vec{B} = -\hat{i} + 5\hat{j} + 5\hat{k} \]

### Step 3: Calculate the magnitude of \(\vec{A} \times \vec{B}\)
\[ |\vec{A} \times \vec{B}| = \sqrt{(-1)^2 + 5^2 + 5^2} = \sqrt{1 + 25 + 25} = \sqrt{51} \]

### Step 4: Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\)
1. Magnitude of \(\vec{A}\):
\[ |\vec{A}| = \sqrt{5^2 + (-3)^2 + 4^2} = \sqrt{25 + 9 + 16} = \sqrt{50} \]

2. Magnitude of \(\vec{B}\):
\[ |\vec{B}| = \sqrt{0^2 + 1^2 + (-1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2} \]

### Step 5: Substitute values into the sine formula
Now substituting the values into the formula for \(\sin \theta\):
\[ \sin \theta = \frac{|\vec{A} \times \vec{B}|}{|\vec{A}| |\vec{B}|} = \frac{\sqrt{51}}{\sqrt{50} \cdot \sqrt{2}} = \frac{\sqrt{51}}{\sqrt{100}} = \frac{\sqrt{51}}{10} \]

### Final Answer
Thus, the sine of the angle between \(\vec{A}\) and \(\vec{B}\) is:
\[ \sin \theta = \frac{\sqrt{51}}{10} \approx 0.7141 \]