# NCERT Solutions for Class 12 Maths Vector Algebra

Hi Students, Welcome to **Amans Maths Blogs (AMB)**. In this post, you will get the * NCERT Solutions for Class 12 Maths Vector Algebra Exercise 10.3*.

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**CBSE NCERT Solutions for Class 12**^{th}Maths**can be downloaded in PDF file. The downloading link is given at last.**

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**CBSE Class 12**^{th}MathsNote: In this solution, the vector is represented by **BOLD** font. For example: **a**, **b,** **OP**, **AB, i, j, k** represent the vectors .

## NCERT Solutions for Class 12 Maths Vector Algebra

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 1.**

Find the angle between two vectors **a** and **b** with magnitudes √3 and 2 respectively having **a**.**b** = √6.

**NCERT Solutions:**

Given that |**a**| = √3 and |**b**| = 2 and **a**.**b** = √6.

Then, the angle between **a** and **b** is

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 2.**

Find the angle between the vectors **i** – 2**j** + 3**k** and 3**i** – 2**j** + **k**.

**NCERT Solutions:**

Given that **a** = **i** – 2**j** + 3**k** and **b** = 3**i** – 2**j** + **k**.

Then,

**a.b** = (**i** – 2**j** + 3**k**).(3**i** – 2**j** + **k**) = 3 + 4 + 3 = 10

Then, the angle between the given vectors **a** and **b** is

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 3.**

Find the projection of the vector **i** – **j** on the vector **i** + **j**.

**NCERT Solutions:**

Given that **a** = **i** – **j** and **b** = **i** + **j**.

Then, |b| =

**a.b** = (**i** – **j**).(**i** + **j**) = 1 – 1 = 0.

Thus, the projection of **a** on **b** is

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 4.**

Find the projection of the vector **i** + 3**j** + 7**k** on the vector 7**i** – **j** + 8**k**.

**NCERT Solutions:**

Given that **a** = **i** + 3**j** + 7**k** and **b** = 7**i** – **j** + 8**k**.

Then, |b| =

**a.b** = (**i** + 3**j** + 7**k**).(7**i** – **j** + 8**k**) = 7 – 3 + 56 = 60.

Thus, the projection of **a** on **b** is

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 5.**

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

**NCERT Solutions:**

Given that .

Thus, these vectors **a**, **b** and **c** are unit vectors.

Now,

Hence, **a** and **b** are perpendicular vectors.

Hence, **b** and **c** are perpendicular vectors.

Hence, **c** and **a** are perpendicular vectors.

Therefore, the vectors **a**, **b** and **c** are mutually vectors.

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 6.**

Find |**a**| and |**b**|, if (**a** + **b**).(**a** – **b**) = 8 and |**a**| = 8|**b**|.

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 7.**

Evaluate the product (3**a** – 5**b**).(2**a** + 7**b**).

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 8.**

Find the magnitude of two vectors **a** and **b**, having the same magnitude and such that the angle between them is 60^{o} and their scalar product is 1/2.

**NCERT Solutions:**

Given that |**a**| = |**b**|, θ = 60^{o} and **a**.**b** = 1/2.

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 9.**

Find |**x**|, if for a unit vector **a**, (**x** – **a**).(**x** + **a**) = 12.

**NCERT Solutions:**

Given that |**a**| = 1

(**x** – **a**).(**x** + **a**) = 12.

⇒ |**x**|^{2} – |**a**|^{2} = 12

⇒ |**x**|^{2} – (1)^{2} = 12

⇒ |**x**|^{2} – 1 = 12

⇒ |**x**|^{2} = 13

⇒ |**x**| = √13

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 10.**

If **a** = 2**i** + 2**j** + 3**k**, **b** = –**i** + 2**j** + **k** and **c** = 3**i** + **j** are such that **a** + λ**b** is perpendicular to **c**, then find the value of **λ**.

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 11.**

Show that |**a**|**b** + |**b**|**a** is perpendicular to |**a**|**b** – |**b**|**a**, for any two nonzero vectors **a** and **b**.

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 12.**

If **a**⋅**a** = 0 and **a**⋅**b** = 0, then what can be concluded about the vector **b**?

**NCERT Solutions:**

Given that **a**⋅**a** = 0 and **a**⋅**b** = 0. It is clear that **a** = 0 (null vector) and **b** is any vector.

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 13.**

If **a**, **b**, **c** are unit vectors such that **a** + **b** + **c** = 0, find the value of **a**.**b** + **b**.**c** + **c**.**a**.

**NCERT Solutions:**

Given that |**a**| = |**b**| = |**c**| = 1 and

**a** + **b** + **c** = 0

⇒ (**a** + **b** + **c**)^{2} = 0

⇒ |**a**|^{2} + |**b**|^{2} + |**c**|^{2} + 2(**a**.**b** + **b**.**c** + **c**.**a**) = 0

⇒ (1)^{2} + (1)^{2} + (1)^{2} + 2(**a**.**b** + **b**.**c** + **c**.**a**) = 0

⇒ 3 + 2(**a**.**b** + **b**.**c** + **c**.**a**) = 0

⇒ (**a**.**b** + **b**.**c** + **c**.**a**) = – 3/2.

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 14.**

If either vector **a** = 0 or **b** = 0, then **a**.**b** = 0. But the converse need not be true. Justify your answer with an example.

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 15.**

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectors **BA **and **BC**].

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 16.**

Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 17.**

Show that the vectors 2**i** − **j** + **k**, **i** − 3**j** − 5**k** and 3**i** − 4**j** − 4**k** form the vertices of a right angled triangle.

**NCERT Solutions:**

**NCERT Solutions for Class 12 Maths Vector Algebra ****Exercise**** 10.3: Ques No 18.**

If **a** is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λ**a** is unit vector if

(A) λ = 1

(B) λ = – 1

(C) a = |λ|

(D) a = 1/|λ|

**NCERT Solutions:**

(D) **a** is a non-zero vector of magnitude a. |**a**| = a

Since λ**a** is a unit vector, then |λ**a**| = 1