A line segment has length 63 and direction ratios
are `3, -2, 6.` The components of the line vector are

Options

A. `-27,18,54`
B. `27,-18,54`
C. `27,-18,-54`
D. `-27,-18,-54`

Explanation

To find the components of the line vector given the length and direction ratios, we can follow these steps:

### Step 1: Understand the Direction Ratios
The direction ratios given are \(3, -2, 6\). These ratios indicate the proportional relationship between the components of the line vector.

### Step 2: Set Up the Proportionality
Let the components of the line vector be represented as \(x, y, z\). We can express these components in terms of a parameter \(k\):
\[ \frac{x}{3} = \frac{y}{-2} = \frac{z}{6} = k \]
From this, we can express \(x, y, z\) as:
\[ x = 3k, \quad y = -2k, \quad z = 6k \]

### Step 3: Use the Length of the Line Segment
The length of the line segment is given as \(63\). The length of the vector can be calculated using the formula:
\[ \sqrt{x^2 + y^2 + z^2} = 63 \]
Substituting the values of \(x, y, z\):
\[ \sqrt{(3k)^2 + (-2k)^2 + (6k)^2} = 63 \]

### Step 4: Simplify the Equation
Squaring both sides gives:
\[ (3k)^2 + (-2k)^2 + (6k)^2 = 63^2 \]
Calculating each term:
\[ 9k^2 + 4k^2 + 36k^2 = 3969 \]
Combining the terms:
\[ 49k^2 = 3969 \]

### Step 5: Solve for \(k\)
Dividing both sides by \(49\):
\[ k^2 = \frac{3969}{49} \]
Calculating the right side:
\[ k^2 = 81 \quad \Rightarrow \quad k = \pm 9 \]

### Step 6: Find the Components
Now substituting \(k\) back into the equations for \(x, y, z\):
1. For \(k = 9\):
\[ x = 3(9) = 27, \quad y = -2(9) = -18, \quad z = 6(9) = 54 \]
Thus, one set of components is \((27, -18, 54)\).

2. For \(k = -9\):
\[ x = 3(-9) = -27, \quad y = -2(-9) = 18, \quad z = 6(-9) = -54 \]
Thus, the other set of components is \((-27, 18, -54)\).

### Final Answer
The components of the line vector are:
\[ (27, -18, 54) \quad \text{and} \quad (-27, 18, -54) \]

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