The projection of line joining (3, 4, 5) and (4, 6, 3) on the line joining (–1, 2, 4) and (1, 0, 5) is

Options

A. `(4)/(3)`
B. `(2)/(3)`
C. `(8)/(3)`
D. `(1)/(3)`

Answer

To find the projection of the line joining the points (3, 4, 5) and (4, 6, 3) on the line joining the points (–1, 2, 4) and (1, 0, 5), we will follow these steps:

### Step 1: Determine the vectors for the lines
Let point A = (3, 4, 5) and point B = (4, 6, 3). The vector AB can be calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} = (4 - 3) \hat{i} + (6 - 4) \hat{j} + (3 - 5) \hat{k} = \hat{i} + 2\hat{j} - 2\hat{k} \]

Let point P = (–1, 2, 4) and point Q = (1, 0, 5). The vector PQ can be calculated as:
\[ \vec{PQ} = \vec{Q} - \vec{P} = (1 - (-1)) \hat{i} + (0 - 2) \hat{j} + (5 - 4) \hat{k} = 2\hat{i} - 2\hat{j} + \hat{k} \]

### Step 2: Calculate the dot product of the vectors
Next, we need to compute the dot product of \(\vec{AB}\) and \(\vec{PQ}\):
\[ \vec{AB} \cdot \vec{PQ} = (1)(2) + (2)(-2) + (-2)(1) = 2 - 4 - 2 = -4 \]

### Step 3: Calculate the magnitude of vector PQ
Now, we calculate the magnitude of vector \(\vec{PQ}\):
\[ |\vec{PQ}| = \sqrt{(2)^2 + (-2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 4: Calculate the projection of AB onto PQ
The projection of vector \(\vec{AB}\) onto vector \(\vec{PQ}\) is given by the formula:
\[ \text{Projection of } \vec{AB} \text{ on } \vec{PQ} = \frac{\vec{AB} \cdot \vec{PQ}}{|\vec{PQ}|^2} \vec{PQ} \]
First, we need \( |\vec{PQ}|^2 \):
\[ |\vec{PQ}|^2 = 3^2 = 9 \]
Now substituting the values:
\[ \text{Projection of } \vec{AB} \text{ on } \vec{PQ} = \frac{-4}{9} \vec{PQ} = \frac{-4}{9}(2\hat{i} - 2\hat{j} + \hat{k}) = \left(-\frac{8}{9}\hat{i} + \frac{8}{9}\hat{j} - \frac{4}{9}\hat{k}\right) \]

### Final Result
The projection of the line joining (3, 4, 5) and (4, 6, 3) on the line joining (–1, 2, 4) and (1, 0, 5) is:
\[ -\frac{8}{9}\hat{i} + \frac{8}{9}\hat{j} - \frac{4}{9}\hat{k} \]