338. Točka $$ T $$ leži na premici skozi točki $$ A(1,0,-3) $$ in $$ B(4,-9,3) $$ tako, da točka $$ B $$ leži med točkama $$ A $$ in $$ T $$ ter je $$ |A T|:|A B|=7: 3 $$. Zapišite koordi- nate točke $$ T $$.

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To find the coordinates of point $$ T $$ on the line passing through points $$ A(1, 0, -3) $$ and $$ B(4, -9, 3) $$, given that $$ B $$ lies between $$ A $$ and $$ T $$ and the ratio $$ |AT| : |AB| = 7 : 3 $$, follow these steps:

1. **Determine the Direction Vector:**
   The direction vector from $$ A $$ to $$ B $$ is:
   $$
   \vec{AB} = B - A = (4 - 1, -9 - 0, 3 - (-3)) = (3, -9, 6)
   $$

2. **Express $$ T $$ in Terms of $$ A $$ and $$ \vec{AB} $$:**
   Since $$ B $$ lies between $$ A $$ and $$ T $$, we can write $$ T $$ as:
   $$
   T = A + t \cdot \vec{AB}
   $$
   where $$ t $$ is a scalar multiplier.

3. **Determine the Scalar $$ t $$:**
   Given the ratio $$ |AT| : |AB| = 7 : 3 $$, we have:
   $$
   \frac{|AT|}{|AB|} = \frac{7}{3} \quad \Rightarrow \quad |AT| = \frac{7}{3} |AB|
   $$
   Since $$ B $$ is between $$ A $$ and $$ T $$, $$ |AT| = |AB| + |BT| $$. Therefore:
   $$
   |AB| + |BT| = \frac{7}{3} |AB| \quad \Rightarrow \quad |BT| = \frac{4}{3} |AB|
   $$
   This implies $$ t = \frac{7}{3} $$.

4. **Calculate the Coordinates of $$ T $$:**
   $$
   T = A + \frac{7}{3} \cdot \vec{AB} = (1, 0, -3) + \frac{7}{3} (3, -9, 6) = (1 + 7, 0 - 21, -3 + 14) = (8, -21, 11)
   $$

**Final Answer:**
$$
T = (8,\ -21,\ 11)
$$
The coordinates of point $$ T $$ are $$ (8, -21, 11) $$.

338. Točka \( T \) leži na premici skozi točki \( A(1,0,-3) \) in \( B(4,-9,3) \) tako, da točka \( B \) leži med točkama \( A \) in \( T \) ter je \( |A T|:|A B|=7: 3 \). Zapišite koordi- nate točke \( T \).

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