Answer

Verified

434.4k+ views

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 \[\overrightarrow {OA} - \overrightarrow {OP} \] .

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 \[\overrightarrow {OA} - \overrightarrow {OP} \] .

Recently Updated Pages

Assertion The resistivity of a semiconductor increases class 13 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

What is pollution? How many types of pollution? Define it