What is the projection of the line segment joining `A(1,7,-5)` and `B(-3,4,-2)` on `y-`axis?

Options

A. 5
B. 4
C. 3
D. 2

Answer

To find the projection of the line segment joining points \( A(1, 7, -5) \) and \( B(-3, 4, -2) \) on the \( y \)-axis, we can follow these steps:

### Step 1: Identify the coordinates of points A and B
The coordinates of point \( A \) are \( (1, 7, -5) \) and the coordinates of point \( B \) are \( (-3, 4, -2) \).

### Step 2: Determine the direction cosines for the \( y \)-axis
The direction cosines for the \( y \)-axis are \( (0, 1, 0) \). This means that the projection will only consider the changes in the \( y \)-coordinate.

### Step 3: Calculate the differences in coordinates
We need to find the differences in the coordinates of points \( A \) and \( B \):
- \( x_2 - x_1 = -3 - 1 = -4 \)
- \( y_2 - y_1 = 4 - 7 = -3 \)
- \( z_2 - z_1 = -2 - (-5) = 3 \)

### Step 4: Use the projection formula
The projection of the line segment \( AB \) on the \( y \)-axis can be calculated using the formula:
\[ \text{Projection} = L(x_2 - x_1) + M(y_2 - y_1) + N(z_2 - z_1) \]
Substituting the values:
- \( L = 0 \)
- \( M = 1 \)
- \( N = 0 \)

Thus, we have:
\[ \text{Projection} = 0 \cdot (-4) + 1 \cdot (-3) + 0 \cdot 3 = 0 - 3 + 0 = -3 \]

### Step 5: Take the absolute value
Since projection is typically expressed as a positive value, we take the absolute value:
\[ |\text{Projection}| = |-3| = 3 \]

### Final Answer
The projection of the line segment joining points \( A(1, 7, -5) \) and \( B(-3, 4, -2) \) on the \( y \)-axis is \( 3 \) units.

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