Given that, the coordinates of two points P and Q are (2, 3, 5) 𝑎𝑛𝑑 (6, 𝑘 − 1, 15)

Question

Given that, the coordinates of two points P and Q are (2, 3, 5) 𝑎𝑛𝑑 (6, 𝑘 − 1, 15)

respectively and that their position vectors are parallel, Calculate the value of |𝑷𝑸|. (4marks)

Answer 1

To find the value of |𝑷𝑸|, we need to find the distance between points P and Q.

The position vector from P to Q is given by:
𝑃𝑄 = (6 - 2, 𝑘 − 1 - 3, 15 - 5)
= (4, 𝑘 - 4, 10)

Since the position vectors are parallel, the direction ratios of the vector 𝑃𝑄 will be proportional to the coordinates of 𝑃.

So, we can write:
4/2 = (𝑘 - 4)/3
Cross multiplying, we get:
2(𝑘 - 4) = 4(3)
2𝑘 - 8 = 12
2𝑘 = 20
𝑘 = 10

Now, we can substitute the value of 𝑘 back into the position vector 𝑃𝑄:
𝑃𝑄 = (4, 𝑘 - 4, 10)
= (4, 10 - 4, 10)
= (4, 6, 10)

The distance between points P and Q, |𝑷𝑸|, can be found using the distance formula:
|𝑷𝑸| = √((6 - 4)^2 + (10 - 6)^2 + (10 - 5)^2)
= √(2^2 + 4^2 + 5^2)
= √(4 + 16 + 25)
= √45

Therefore, the value of |𝑷𝑸| is √45.

Answer 2

To calculate the value of |𝑷𝑸| (the distance between points P and Q), we can use the distance formula in three-dimensional space.

The distance formula in three dimensions is given by:

|𝑿2−𝑿1| = √((𝑥2−𝑥1)² + (𝑦2−𝑦1)² + (𝑧2−𝑧1)²)

Let's plug in the coordinates for points P and Q into the distance formula:

|𝑷𝑸| = √((6−2)² + ((𝑘−1)−3)² + (15−5)²)

Simplifying inside the square root:

|𝑷𝑸| = √(4² + (𝑘−4)² + 10²)

|𝑷𝑸| = √(16 + (𝑘−4)² + 100)

|𝑷𝑸| = √(116 + (𝑘−4)²)

We know that the position vectors of points P and Q are parallel, which means they have the same direction. Therefore, their differences in x, y, and z coordinates must be proportional.

Comparing the x-coordinates:

6−2 = 4

Comparing the y-coordinates:

𝑘−1−3 = 𝑘−4

Comparing the z-coordinates:

15−5 = 10

Since the ratios of the differences in coordinates are constant, we can solve for 𝑘 by equating any two ratios and solving for 𝑘. Let's equate the ratio of the x-coordinates to the ratio of the z-coordinates:

4/10 = 4/(15−5)

Simplifying:

4/10 = 4/10

This equation is true for any value of 𝑘. Therefore, the value of 𝑘 does not matter in this case. The value of |𝑷𝑸| remains the same regardless of the value of 𝑘.

Thus, the value of |𝑷𝑸| is √(116 + (𝑘−4)²).