**14. For given vectors, ****\(\vec a = 2\hat i - \hat j + 2\hat k\)** and **\(\vec b= -\hat i + \hat j-\hat k\)**, find the unit vector in the direction of the vector **\(\vec a + \vec b\)**.

**Answer:**

The given vectors are \(\vec a = 2\hat i - \hat j + 2\hat k\) and \(\vec b= -\hat i + \hat j-\hat k\)

Hence, the unit vector in the direction of **\((\vec a + \vec b)\)** is

**15. Find a vector in the direction of vector ****\(5\hat i - \hat j + 2\hat k\)** which has magnitude 8 units.

**Answer:**

Hence, the vector in the direction of vector **\(5\hat i - \hat j + 2\hat k\)** which has magnitude 8 units is given by,

**16. Show that the vectors ****\( 2\hat i - 3\hat j + 4\hat k\)** and **\(-4\hat i + 6\hat j-8\hat k\)** are collinear.

**Answer:**

Hence, the given vectors are collinear.

**17. Find the direction cosines of the vector ****\( \hat i + 2\hat j + 3\hat k\)**.

**Answer:**

Hence, the direction cosines of

**18. Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.**

**Answer:**

The given points are A (1, 2, –3) and B (–1, –2, 1).

Hence, the direction cosines of \(\vec{AB}\) are

**19. Show that the vector \(\hat i + \hat j + \hat k\) is equally inclined to the axes OX, OY, and OZ.**

**Answer:**

Let **\(\vec a=\hat i + \hat j + \hat k\) **

Then,

**\(|\vec a| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt3\)**

Therefore, the direction cosines of

Now, let α, β, and γ be the angles formed by \(\vec a\) with the positive directions of x, y, and z axes.

Then, we have

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

**20. Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).**

**Answer:**

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

**21. Show that the points A, B and C with position vectors,**

\(\vec a = 3\hat i - 4\hat j - 4\hat k, \vec b= 2\hat i - \hat j + \hat k \;and \; \vec c = \hat i - 3\hat j -5\hat k \)

**respectively form the vertices of a right angled triangle.**

**Answer:**

Position vectors of points A, B, and C are respectively given as:

Hence, ABC is a right-angled triangle.

**22. In triangle ABC which of the following is not true:**

**Answer:**

On applying the triangle law of addition in the given triangle, we have:

From equations (1) and (3), we have:

Hence, the equation given in alternative C is incorrect.

The correct answer is C.

**23. If \(\vec a\) and \(\vec b\) are two collinear vectors, then which of the following are incorrect: **

**A. \(\vec b = \lambda\vec a\), for some scalar λ **

**B. \(\vec a = \pm \vec b\)**

**C. the respective components of \(\vec a\) and \(\vec b\) are proportional**

**D. both the vectors \(\vec a\) and \(\vec b\) have same direction, but different magnitudes**

**Answer:**

If ** ****\(\vec a\)** and **\(\vec b\)**are two collinear vectors, then they are parallel.

Therefore, we have:

**\(\vec b = \lambda\vec a\)**(For some scalar λ)

Thus, the respective components of ** ****\(\vec a\)** and **\(\vec b\)** are proportional.

However, vectors** ****\(\vec a\)** and **\(\vec b\)** can have different directions.

Hence, the statement given in D is incorrect.

The correct answer is D.

**24. Find the angle between two vectors \(\vec a\) and \(\vec b\) with magnitudes and 2, respectively having \(\vec a.\vec b = \sqrt 6\).**

**Answer:**

It is given that,

Hence, the angle between the given vectors **\(\vec a\)** and **\(\vec b\) **is **\(\frac\pi4.\)**