The distance between two points P and Q is d and the length of their projections of PQ on the co-ordinate planes are `d _(1),d_(2), d_(3).` Then `d _(1) ^(2) + d_(2)^(2) + d_(3) ^(2) = kd ^(2),` where k is `"_____."`

Answer

To solve the problem, we need to establish the relationship between the distance \( d \) between points \( P \) and \( Q \) and the projections of the line segment \( PQ \) on the coordinate planes.

Let’s denote the coordinates of points \( P \) and \( Q \) as follows:
- \( P(x_1, y_1, z_1) \)
- \( Q(x_2, y_2, z_2) \)

### Step 1: Calculate the distance \( d \) between points \( P \) and \( Q \)

The distance \( d \) can be calculated using the distance formula in three-dimensional space:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 2: Define the projections \( d_1, d_2, d_3 \)

The projections of the line segment \( PQ \) on the coordinate planes are defined as:
- \( d_1 \): Projection on the yz-plane
- \( d_2 \): Projection on the xz-plane
- \( d_3 \): Projection on the xy-plane

These projections can be expressed as:
- \( d_1 = \sqrt{(y_2 - y_1)^2 + (z_2 - z_1)^2} \)
- \( d_2 = \sqrt{(x_2 - x_1)^2 + (z_2 - z_1)^2} \)
- \( d_3 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

### Step 3: Establish the relationship between the projections and the distance

We need to find the relationship given by:

\[ d_1^2 + d_2^2 + d_3^2 = k d^2 \]

Calculating \( d_1^2, d_2^2, d_3^2 \):

\[ d_1^2 = (y_2 - y_1)^2 + (z_2 - z_1)^2 \]
\[ d_2^2 = (x_2 - x_1)^2 + (z_2 - z_1)^2 \]
\[ d_3^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]

### Step 4: Add the squares of the projections

Now, we add these equations:

\[ d_1^2 + d_2^2 + d_3^2 = \left[(y_2 - y_1)^2 + (z_2 - z_1)^2\right] + \left[(x_2 - x_1)^2 + (z_2 - z_1)^2\right] + \left[(x_2 - x_1)^2 + (y_2 - y_1)^2\right] \]

Simplifying this gives:

\[ d_1^2 + d_2^2 + d_3^2 = 2(x_2 - x_1)^2 + 2(y_2 - y_1)^2 + 2(z_2 - z_1)^2 \]

### Step 5: Relate it to \( d^2 \)

Now, we can express \( d^2 \):

\[ d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 \]

Thus, we can rewrite:

\[ d_1^2 + d_2^2 + d_3^2 = 2[(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2] = 2d^2 \]

### Conclusion

From the above steps, we find that:

\[ d_1^2 + d_2^2 + d_3^2 = 2d^2 \]

Thus, the value of \( k \) is:

\[ k = 2 \]