The projection of a line segment on the axis 2, 3, 6 respectively. Then find the length of line segment.

Options

A. `7`
B. `5`
C. `1`
D. `11`

Answer

To find the length of the line segment given its projections on the axes, we can follow these steps:

### Step 1: Understand the Projections
The projections of the line segment on the x-axis, y-axis, and z-axis are given as 2, 3, and 6 respectively. This means that if we denote the line segment as AB, the components of the vector AB can be represented as:
- \( AB_x = 2 \) (projection on the x-axis)
- \( AB_y = 3 \) (projection on the y-axis)
- \( AB_z = 6 \) (projection on the z-axis)

### Step 2: Represent the Line Segment as a Vector
We can represent the line segment AB as a vector in three-dimensional space:
\[ \vec{AB} = 2\hat{i} + 3\hat{j} + 6\hat{k} \]

### Step 3: Use the Distance Formula
To find the length of the line segment AB, we use the distance formula in three dimensions:
\[ \text{Length of } AB = |\vec{AB}| = \sqrt{(AB_x)^2 + (AB_y)^2 + (AB_z)^2} \]

### Step 4: Substitute the Values
Now, substituting the values of the projections into the formula:
\[ |\vec{AB}| = \sqrt{(2)^2 + (3)^2 + (6)^2} \]

### Step 5: Calculate Each Term
Calculating each term:
\[ |\vec{AB}| = \sqrt{4 + 9 + 36} \]

### Step 6: Sum the Values
Now, sum the values inside the square root:
\[ |\vec{AB}| = \sqrt{49} \]

### Step 7: Find the Square Root
Finally, take the square root:
\[ |\vec{AB}| = 7 \]

### Conclusion
The length of the line segment AB is 7 units.

---