Find the angle between the vectors `hat(i)+3hat(j)+7hat(k) and 7hat(i)-hat(j)+8hat(k)`.

Answer

To find the angle between the vectors \(\hat{i} + 3\hat{j} + 7\hat{k}\) and \(7\hat{i} - \hat{j} + 8\hat{k}\), we can follow these steps:

### Step 1: Define the Vectors
Let
\[ \mathbf{A} = \hat{i} + 3\hat{j} + 7\hat{k} \]
and
\[ \mathbf{B} = 7\hat{i} - \hat{j} + 8\hat{k} \]

### Step 2: Calculate the Dot Product
The dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \]
Calculating the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (1)(7) + (3)(-1) + (7)(8) \]
\[ = 7 - 3 + 56 \]
\[ = 60 \]

### Step 3: Calculate the Magnitudes of the Vectors
The magnitude of vector \(\mathbf{A}\) is:
\[ |\mathbf{A}| = \sqrt{1^2 + 3^2 + 7^2} = \sqrt{1 + 9 + 49} = \sqrt{59} \]

The magnitude of vector \(\mathbf{B}\) is:
\[ |\mathbf{B}| = \sqrt{7^2 + (-1)^2 + 8^2} = \sqrt{49 + 1 + 64} = \sqrt{114} \]

### Step 4: Use the Dot Product to Find the Angle
The formula relating the dot product and the angle \(\theta\) between two vectors is:
\[ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \]
Substituting the values we calculated:
\[ 60 = \sqrt{59} \cdot \sqrt{114} \cdot \cos \theta \]

### Step 5: Solve for \(\cos \theta\)
Rearranging gives:
\[ \cos \theta = \frac{60}{\sqrt{59} \cdot \sqrt{114}} \]

### Step 6: Calculate \(\theta\)
To find \(\theta\), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{60}{\sqrt{59 \cdot 114}}\right) \]

### Final Answer
Thus, the angle between the vectors \(\hat{i} + 3\hat{j} + 7\hat{k}\) and \(7\hat{i} - \hat{j} + 8\hat{k}\) is:
\[ \theta = \cos^{-1}\left(\frac{60}{\sqrt{6726}}\right) \]