 Questions & Answers    Question Answers

# Using vectors show that the point A (-2, 3, 5), B (7, 0, -1), C (-3, -2, -5) and D (3, 4, 7) are such that AB and CD intersect at the point P (1, 2, 3).  Answer Verified
Hint: Show that the point P lies on AB by showing that AB and AP are collinear. Then show that the point P lies on CD by showing that CP and CD are collinear. Since P lies on both AB and CD, it is the point of intersection of AB and CD.

Complete step-by-step answer:
From the given points we calculate the position vectors of each point from origin as follows:
$\overrightarrow {OA} = - 2i + 3j + 5k$
$\overrightarrow {OB} = 7i - 1k$
$\overrightarrow {OC} = - 3i - 2j - 5k$
$\overrightarrow {OD} = 3i + 4j + 7k$
$\overrightarrow {OP} = i + 2j + 3k$

We now find the vector $\overrightarrow {AP}$ as follows:
$\overrightarrow {AP} = \overrightarrow {OP} - \overrightarrow {OA}$
Substituting the vectors, we get:
$\overrightarrow {AP} = (i + 2j + 3k) - ( - 2i + 3j + 5k)$
Simplifying, we get:
$\overrightarrow {AP} = 3i - j - 2k........(1)$

Now, we find the vector $\overrightarrow {AB}$ as follows:
$\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA}$
Substituting the vectors, we get:
$\overrightarrow {AB} = (7i - 1k) - ( - 2i + 3j + 5k)$
Simplifying the expression, we get:
$\overrightarrow {AB} = 9i - 3j - 6k.........(2)$

Comparing equation (1) and equation (2), we observe:
$\overrightarrow {AB} = 3\overrightarrow {AP}$
Hence, the point P lies on the line AB.
We now find the vector $\overrightarrow {CP}$ as follows:
$\overrightarrow {CP} = \overrightarrow {OP} - \overrightarrow {OC}$
Substituting the vectors, we get:
$\overrightarrow {CP} = (i + 2j + 3k) - ( - 3i - 2j - 5k)$
Simplifying, we get:
$\overrightarrow {CP} = 4i + 4j + 8k........(3)$

Now, we find the vector $\overrightarrow {CD}$ as follows:
$\overrightarrow {CD} = \overrightarrow {OD} - \overrightarrow {OC}$
Substituting the vectors, we get:
$\overrightarrow {CD} = (3i + 4j + 7k) - ( - 3i - 2j - 5k)$
Simplifying the expression, we get:
$\overrightarrow {CD} = 6i + 6j + 12k.........(4)$
Comparing equation (3) and (4), we observe:
$\overrightarrow {CD} = \dfrac{3}{2}\overrightarrow {CP}$

Hence, the point P lies on the line CD.
Since, P lies on both the lines AB and CD, it is the point of intersection of the two lines.

Hence, we showed that AB and CD intersect at point P.

Note: The way we are asked to solve is clearly mentioned as using vectors, it is an error to solve using any other method other than vector method. Also, vector $\overrightarrow {AP}$ is $\overrightarrow {OP} - \overrightarrow {OA}$ and not $\overrightarrow {OA} - \overrightarrow {OP}$ .
Bookmark added to your notes.
View Notes
To Find the Weight of a Given Body Using Parallelogram Law of Vectors  Coordinate Geometry  Vectors  Maths Coordinate Geometry Formulas  Two Dimensional Coordinate Geometry  Coplanar Vectors  Coordinate Geometry For Class 10  Addition of Vectors  Types of Vector  Coplanarity of Vectors  