Find the length of the lines segment joining the points whose position vectors are `7bar(j)+10bar(k),-4bar(i)+9bar(j)+6bar(k)`.

Answer

To find the length of the line segment joining the points whose position vectors are \( \vec{A} = 7\hat{j} + 10\hat{k} \) and \( \vec{B} = -4\hat{i} + 9\hat{j} + 6\hat{k} \), we can follow these steps:

### Step 1: Write down the position vectors
The position vectors are given as:
- \( \vec{A} = 0\hat{i} + 7\hat{j} + 10\hat{k} \)
- \( \vec{B} = -4\hat{i} + 9\hat{j} + 6\hat{k} \)

### Step 2: Find the vector joining the points
The vector \( \vec{AB} \) joining points \( A \) and \( B \) can be calculated using the formula:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AB} = (-4\hat{i} + 9\hat{j} + 6\hat{k}) - (0\hat{i} + 7\hat{j} + 10\hat{k}) \]
This simplifies to:
\[ \vec{AB} = -4\hat{i} + (9 - 7)\hat{j} + (6 - 10)\hat{k} \]
\[ \vec{AB} = -4\hat{i} + 2\hat{j} - 4\hat{k} \]

### Step 3: Calculate the length of the vector
The length of the vector \( \vec{AB} \) is given by the formula:
\[ |\vec{AB}| = \sqrt{x^2 + y^2 + z^2} \]
where \( x, y, z \) are the coefficients of \( \hat{i}, \hat{j}, \hat{k} \) respectively. Here, \( x = -4 \), \( y = 2 \), and \( z = -4 \).

Substituting these values:
\[ |\vec{AB}| = \sqrt{(-4)^2 + (2)^2 + (-4)^2} \]
Calculating each term:
\[ |\vec{AB}| = \sqrt{16 + 4 + 16} \]
\[ |\vec{AB}| = \sqrt{36} \]
\[ |\vec{AB}| = 6 \]

### Final Answer
The length of the line segment joining the points is \( 6 \).

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