Certainly! To find the coordinates of point [tex]\( D \)[/tex] on the line segment [tex]\( AB \)[/tex], such that the ratio [tex]\( AD:DB \)[/tex] is [tex]\(2:1\)[/tex], we will use the section formula. Let's go through this step-by-step:
### Step 1: Identify the position vectors of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
From the problem, we have:
- The position vector of point [tex]\( A \)[/tex] is [tex]\( \mathbf{a} = 4i + 4j - 6k \)[/tex].
- The position vector of point [tex]\( B \)[/tex] is [tex]\( \mathbf{b} = 10i + 4j + 12k \)[/tex].
### Step 2: Understand the Ratio
Given that [tex]\( AD:DB \)[/tex] is [tex]\(2:1\)[/tex], this means:
- [tex]\( m = 2 \)[/tex] (the part of the line segment from [tex]\( A \)[/tex] to [tex]\( D \)[/tex])
- [tex]\( n = 1 \)[/tex] (the part of the line segment from [tex]\( D \)[/tex] to [tex]\( B \)[/tex])
### Step 3: Apply the Section Formula
The section formula in vectors states that a point [tex]\( D \)[/tex] dividing the line segment [tex]\( AB \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ \mathbf{D} = \frac{n \mathbf{a} + m \mathbf{b}}{m+n} \][/tex]
Substitute [tex]\( \mathbf{a} \)[/tex], [tex]\( \mathbf{b} \)[/tex], [tex]\( m \)[/tex], and [tex]\( n \)[/tex] into the formula:
[tex]\[ \mathbf{D} = \frac{1(4i + 4j - 6k) + 2(10i + 4j + 12k)}{2+1} \][/tex]
### Step 4: Simplify the Expression
Now, simplify the expression:
[tex]\[
\mathbf{D} = \frac{1 \cdot (4i + 4j - 6k) + 2 \cdot (10i + 4j + 12k)}{3}
\][/tex]
[tex]\[
\mathbf{D} = \frac{(4i + 4j - 6k) + (20i + 8j + 24k)}{3}
\][/tex]
[tex]\[
\mathbf{D} = \frac{(4i + 20i) + (4j + 8j) + (-6k + 24k)}{3}
\][/tex]
[tex]\[
\mathbf{D} = \frac{24i + 12j + 18k}{3}
\][/tex]
### Step 5: Divide by the Denominator
Now, divide each component by 3:
[tex]\[
\mathbf{D} = 8i + 4j + 6k
\][/tex]
### Step 6: Identify Coordinates of [tex]\( D \)[/tex]
The coordinates of point [tex]\( D \)[/tex] are:
[tex]\[
\boxed{(8.0, 4.0, 6.0)}
\][/tex]
Thus, the coordinates of the point [tex]\( D \)[/tex] that divides the line segment [tex]\( AB \)[/tex] in the ratio [tex]\( 2:1 \)[/tex] are [tex]\( (8.0, 4.0, 6.0) \)[/tex].
