# Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 10 Vector Algebra.

## Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra

### Plus Two Maths Vector Algebra 3 Marks Important Questions

Question 1.
(i) With help of a suitable figure for any three vectorsa,bandc show that $$(\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\vec{b}+\bar{c})$$
(ii) If $$\bar{a}$$ = i – j + k and $$\bar{b}$$ = 2i – 2j – k. What is the projection of a on b? (March – 2011)
(i) Answered in previous years questions
No. 1(ii) (6 Mark question)
(ii) Projection of $$\bar{a} \text { on } \bar{b}=\frac{\bar{a} \cdot \bar{b}}{|\bar{b}|}=\frac{2+2-1}{\sqrt{4+4+1}}=1$$

Question 2.
(i) If $$\bar{a}$$ = 3i – j – 5k and $$\bar{b}$$ = i – 5j + 3k Show that $$\bar{a}$$ + $$\bar{b}$$ and a bare perpendicular.
(ii) Given the position vectors of three points as A(i – j + k); B(4i + 5j + 7k) C(3i + 3j + 5k)
(a)Find $$\bar{AB}$$ and $$\bar{BC}$$
(b) Prove that A,B and C are collinear points. (March – 2011)

Question 3.
(i) Write the unit vector in direction of i + 2j – 3k.
(ii) If $$\overline{P Q}$$ = 31 + 2j — k and the coordinate of P are(1, -1,2) , find the coordinates of Q. (May – 2012)

Question 4.
(a) The angle between the vectors $$\bar{a}$$ and $$\bar{b}$$ such that $$|\vec{a}|=|\bar{b}|=\sqrt{2}$$
$$\bar{a}$$.$$\bar{b}$$ = 1 is
$$\begin{array}{lll} \text { (i) } \frac{\pi}{2} & \text { (ii) } \frac{\pi}{3} & \text { (iii) } \frac{\pi}{4} & \text { (iv) } 0 \end{array}$$
(b) Find the unit vector along $$\bar{a}-\bar{b}$$ where $$\bar{a}$$ = i + 3j – k and $$\bar{b}$$ 3i + 2j + k (March -2016)

### Plus Two Maths Vector Algebra 4 Marks Important Questions

Question 1.
Consider the vectors $$\bar{a}$$ = 21+ j – 2k and $$\bar{b}$$ = 6i – 3j + 2k.
(i) Find $$\bar{a} \bar{b}$$ and $$\bar{a} \times \bar{b}$$.
(ii) Verity that $$|\bar{a} \times \bar{b}|=|\vec{a}|^{2}|\bar{b}|^{2}-(\bar{a} \cdot \bar{b})^{2}$$ (March – 2012)

Question 2.
(i) For any three vectors $$\bar{a}, \bar{b}, \bar{c}$$, show that $$\bar{a} \times(\bar{b}+\bar{c})+\bar{b} \times(\bar{c}+\bar{a})+\bar{c} \times(\bar{a}+\bar{b})=0$$
(ii) Given A (1, 1, 1), B (1, 2, 3), C (2, 3, 1) are the vertices of MBCa triangle. Find the area of the ∆ABC (May – 2012)

Question 3.
Consider A (2, 3, 4) , B (4, 3, 2) and C (5, 2, -1) be any three points
(i) Find the projection of $$\overline{B C}$$ on $$\overline{A B}$$
(ii) Find the area of triangle ABC (March – 2013)

Question 4.
(i) Find the angle between the vectors $$\bar{a}$$ =3i + 4j + k and $$\bar{b}$$ = 2i + 3j – k
(ii) The adjacent sides of a parallelogram are $$\bar{a}$$ = 3i + λj + 4k and $$\bar{b}$$ = i – λj + k
(a) Find $$\bar{a} \times \bar{b}$$
(b) If the area of the parallelogram is square units, find the value of A (May – 2013)

Question 5.
Let $$\bar{a}$$ = 2i – j + 2k and $$\bar{b}$$ = 6i + 2j + 3k
(i) Find a unit vector in the direction of $$\bar{a}$$ + $$\bar{b}$$
(ii) Find the angle between a and b (March – 2014)

Question 6.
Consider the triangle ABC with vertices A(1, 1, 1) , B (1, 2, 3) and C (2, 3, 1)
(i) Find $$\overline{A B}$$ and $$\overline{A C}$$
(ii) Find $$\overline{A B}$$ x $$\overline{A C}$$
(iii) Hence find the area of the triangle (March – 2014)

Question 7.
Consider the vectors $$\bar{a}$$ = i – 7j + 7k; $$\bar{b}$$ = 3i – 2j + 2k
(a) Find $$\bar{a b}$$.
(b) Find the angle between $$\bar{a}$$ and $$\bar{b}$$.
(c) Find the area of parallelogram with adjacent sides $$\bar{a}$$ and $$\bar{b}$$. (May – 2014)

Question 8.
(a) If the points A and B are (1, 2, -1) and (2, 1, -1) respectively, then is
(i) i + J
(ii) i – J
(iii) 2i + j – k
(iv) i + j + k
(b) Find the value of for which the vectors 2i – 4j + 5k, i – λj + k and 3i + 2j – 5k are coplanar.
(c) Find the angle between the vectors a = 2i + j – k and b = i – j + k (March – 2016)

Question 9.
(i) $$(\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})$$ is equaito
$$\begin{array}{lll} \text { (a) } \bar{a} & \text { (b) }|\bar{a}|^{2}-|\bar{b}|^{2} & \text { (c) } \bar{a} \times \bar{b} \text { (d) } 2(\bar{a} \times \bar{b}) \end{array}$$

(ii) If $$\bar{a}$$ and $$\bar{b}$$ are any two vectors, then
$$(\bar{a} \times \bar{b})^{2}=\left|\begin{array}{ll} \bar{a} \cdot \bar{a} & \bar{a} \cdot \bar{b} \\ \bar{a} \cdot \bar{b} & \bar{b} \bar{b} \end{array}\right|$$

(iii) Using vectors, show that the points A(1, 2, 7), B(2, 6, 3), C(3, 10, -i) are collinear. (May – 2016)

### Plus Two Maths Vector Algebra 6 Marks Important Questions

Question 1.
(i) Find a vector in the direction of $$\bar{r}$$ = 3E – 4j that has a magnitude of 9.
(ii) For any three vectors $$\bar{a,b}$$ and $$\bar{c}$$, and Prove that $$(\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\bar{b}+\bar{c})$$.
(iii) Find a unit vector perpendicular to $$\bar{a}+\bar{b}$$ and $$\bar{a}-\bar{b}$$, where $$\bar{a}$$ = i – 3j + 3k and $$\bar{b}$$ and $$/bar{c}$$ = 3E—3j+2k. (March – 2010)
(i) Unit vector of magnitude 9

Question 2.
Let A(2, 3, 4), B(4, 3, 2) and C(5, 2, -1) be three points
(i) Find $$\overline{A B}$$ and $$\overline{B C}$$
(ii) Find the projection of $$\overline{B C}$$ on $$\overline{A B}$$
(iii) Fiñd the area of the triangle ABC. (May – 2010)

Question 3.
ABCD s a parallelogram with A as the origin, $$\bar{b}$$ and $$\bar{d}$$ are the position vectors of B and D respectively.

(i) What is the position vector of C?
(ii) What is the angle between $$\bar{AB}$$ and $$\bar{AD}$$?
(iii) If $$|\overrightarrow{A C}|=|\overrightarrow{B D}|$$, show that ABCD is a rectangle. (May – 2011)
(i) Since ABCD is a parallelogram with A as the

Question 4.
(a) If $$\bar{a}, \bar{b}, \bar{c}, \bar{d}$$ respectively are the position vectors representing the vertices A,B,C,D of a parallelogram, then write $$/bar{d}$$ in terms of $$\bar{a}, \bar{b}, \bar{c}$$.
(b) Find the projection vector of $$/bar{b}$$ = i + 2j + k along the vector $$/bar{a}$$ = 21 – i – j + 2k. Also write $$/bar{b}$$ as the sum of a vector along $$/bar{a}$$ and a perpendicular to $$/bar{a}$$.
(C) Find the area of a parallelogram for which the vectors 21 + j, 31 + j +4k are adjacent sides. (March – 2015)

Question 5.
(a) Write the magnitude of a vector $$/bar{a}$$ in terms of dot product.
(b) If $$\bar{a}, \bar{b}, \bar{a}+\bar{b}$$ are unit vectors, then prove that the angle between $$/bar{a}$$ and $$/bar{b}$$ is $$\frac{2 \pi}{3}$$
(c) If 2i + j – 3k and mi + 3j – k are perpendicular to each other, then find ‘m’.
Also find the area of the rectangle having these two vectors as sides. (March – 2015)
(a) Find $$\overline{A B}$$ and $$\overline{A C}$$
(b) Find $$\angle A$$