If `p=7hati-2hatj+3hatk and q=3hati+hatj+5hatk`, then find the magnitude of p-2q.

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Define the vectors
We have the vectors:
\[ \mathbf{p} = 7\hat{i} - 2\hat{j} + 3\hat{k} \]
\[ \mathbf{q} = 3\hat{i} + \hat{j} + 5\hat{k} \]

### Step 2: Calculate \(2\mathbf{q}\)
First, we need to calculate \(2\mathbf{q}\):
\[ 2\mathbf{q} = 2(3\hat{i} + \hat{j} + 5\hat{k}) = 6\hat{i} + 2\hat{j} + 10\hat{k} \]

### Step 3: Calculate \(\mathbf{p} - 2\mathbf{q}\)
Now, we will subtract \(2\mathbf{q}\) from \(\mathbf{p}\):
\[ \mathbf{p} - 2\mathbf{q} = (7\hat{i} - 2\hat{j} + 3\hat{k}) - (6\hat{i} + 2\hat{j} + 10\hat{k}) \]
Distributing the negative sign:
\[ = 7\hat{i} - 2\hat{j} + 3\hat{k} - 6\hat{i} - 2\hat{j} - 10\hat{k} \]

### Step 4: Combine like terms
Now, we combine the like terms:
\[ = (7 - 6)\hat{i} + (-2 - 2)\hat{j} + (3 - 10)\hat{k} \]
\[ = 1\hat{i} - 4\hat{j} - 7\hat{k} \]

### Step 5: Find the magnitude of \(\mathbf{p} - 2\mathbf{q}\)
The magnitude of a vector \(\mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) is given by:
\[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
For our vector \(1\hat{i} - 4\hat{j} - 7\hat{k}\):
\[ |\mathbf{p} - 2\mathbf{q}| = \sqrt{1^2 + (-4)^2 + (-7)^2} \]
Calculating the squares:
\[ = \sqrt{1 + 16 + 49} \]
\[ = \sqrt{66} \]

### Final Answer
Thus, the magnitude of \(\mathbf{p} - 2\mathbf{q}\) is:
\[ \sqrt{66} \]

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