# Class 12 RD Sharma Solutions – Chapter 23 Algebra of Vectors – Exercise 23.6 | Set 2

### Question 11: Find the position vector of the mid-point of the vector joining the points P() and Q().

**Solution:**

The mid-point of the line segment joining 2 vectors is given by:

=>

=>

=>

=>

### Question 12: Find the unit vector in the direction of the vector , where P and Q are the points (1,2,3) and (4,5,6).

**Solution:**

Let,

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=>Unit vector is,

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### Question 13: Show that the points A(), B(), C() are the vertices of a right-angled triangle.

**Solution:**

Let,

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The line segments are,

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=>=>

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=>=>

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=>The magnitudes of the sides are,

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=>As we can see that

=>Thus, ABC is a right-angled triangle.

### Question 14: Find the position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).

**Solution:**

Let,

=>

=>

The mid-point of the line segment joining 2 vectors is given by:

=>

=>

=>

=>

### Question 15: Find the value of x for which x() is a unit vector.

**Solution:**

The magnitude of the given vector is,

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For it to be a unit vector,

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### Question 16: If , and , find a unit vector parallel to .

**Solution:**

Given, , and

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Thus, the unit vector is,

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### Question 17: If , and , find a vector of magnitude 6 units which is parallel to the vector .

**Solution:**

Given, , and

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Unit vector in that direction is,

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=>Given that the vector has a magnitude of 6,

=>Required vectors are : =

### Question 18: Find a vector of magnitude 5 units parallel to the resultant of the vector and .

**Solution:**

Given, and

The resultant vector will be given by,

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Unit vector is,

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=>Given that the vector has a magnitude of 5,

=>Required vectors are:

### Question 19: The two vectors and represent the sides and respectively of the triangle ABC. Find the length of the median through A.

**Solution:**

Let D be the point on BC, on which the median through A touches.

D is also the mid-point of BC.

The median is thus given by:

=>

=>

=>

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=>Thus, the length of the median is,

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=>

=>units