Here, we are going to see some word problems with solution in vector.

**Problem 1 :**

If a vector and b vector represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal.

**Solution :**

AB vector = a vector, AC vector = b vector

AB vector + BC vector = AC vector

BC vector = AC vector - AB vector

BC vector = b vector - a vector

The sides AB and CD are in opposite direction.

CD vector = -AB vector = -a vector

CD vector = - a vector

AD vector = b vector - a vector

To find the length of DA, we have to multiply AD vector by negative.

- AD vector = -(b vector - a vector)

DA vector = a vector - b vector

Now let us find the length of other diagonal BD.

AB vector + BD vector = AD vector

BD vector = AD vector - AB vector

= b vector - a vector - a vector

BD vector = b vector - 2a vector

**Problem 2 :**

If PO vector + OQ vector = QO vector +OR vector, prove that the points P, Q, R are collinear.

**Solution :**

**PO vector + OQ vector = QO vector +OR vector**

**-OP vector + OQ vector = -OQ vector + OR vector**

**OQ vector ****- OP vector = ****OR vector ****- OQ vector**

**PQ vector = QR vector**

**Because they are equal, they are parallel and they have a common point Q.**

**So, the points P, Q and R are collinear.**

**Problem 3 :**

If D is the midpoint of the side BC of a triangle ABC, prove that AB vector + AC vector = 2AD vector.

**Solution :**

From the given information, let us draw a rough diagram.

In triangle ABD,

AB vector + BD vector = AD vector

AB vector = AD vector - BD vector ---(1)

In triangle ADC,

AD vector + DC vector = AC vector

AC vector = AD vector + DC vector ----(2)

(1) + (2)

AB vector + AC vector = AD vector - BD vector + AD vector + DC vector

AB vector + AC vector = 2AD vector - BD vector + DC vector. Since BD and DC are in same magnitude, they will get canceled.

AB vector + AC vector = 2AD vector

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