TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Students must practice these TS Inter 1st Year Maths 1A Important Questions Chapter 4 Addition of Vectors to help strengthen their preparations for exams.

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 1.
Show that the points whose position vectors are \(-2 \overline{\mathbf{a}}+3 \overline{\mathbf{b}}+5 \overline{\mathbf{c}}, \overline{\mathbf{a}}+2 \overline{\mathbf{b}}+3 \overline{\mathbf{c}}, 7 \overline{\mathbf{a}}-\overline{\mathbf{c}}\) are collinear when, \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are non coplanar vectors.
Solution:
Let O be the origin of reference so that
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 1

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 2.
If A B C D E F is a regular hexagon with centre o then prove that \(\overline{\mathrm{AB}}+\overline{\mathrm{AC}}+\overline{\mathrm{AD}}+\overline{\mathrm{AE}}+\overline{\mathrm{AF}}= \overline{3 \mathrm{AD}}=6 \overline{\mathrm{AO}}\).
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 3

Question 3.
In the two dimensional plane, prove by using vector methods, the equation of the line whose intercepts on the axes are a and b is \(\frac{x}{a}+\frac{y}{b}\) = 1
Solution:
Let A = (a,0), B = (0,b). P = (x,y)
Let O be the origin so that
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 4

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 4.
Find a unit vector in the direction of the vector = \(\overline{\mathbf{a}}=2 \overline{\mathbf{i}}+3 \overline{\mathbf{j}}+\overline{\mathrm{k}}\)
Solution:
The unit vector in the direction of the vector
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 5

Question 5.
Find a vector in the direction of vector \(\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}\) that has magnitude 7 units.
Solution:
The unit vector in the direction of given
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 6

Question 6.
Find the unit vector in the direction of sum of the vectors \(\overline{\mathbf{a}}=2 \overline{\mathbf{i}}+2 \overline{\mathbf{j}}-5 \overline{\mathbf{k}}\) and \( \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}+3 \overline{\mathbf{k}}\)
Solution:
The sum of the vectors is
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 7

Question 7.
Write the direction ratios of the vector \(\overline{\mathbf{a}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}-\mathbf{2} \overline{\mathbf{k}}\) and hence calculate its direction cosines.
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 8

Question 8.
Consider the two points P and Q with position vectors \(\overline{\mathrm{OP}}=3 \overline{\mathrm{a}}-2 \overline{\mathrm{b}}\) and \(\overline{\mathrm{OQ}}= \overline{\mathbf{a}}+\overline{\mathbf{b}}\). Find the position vector of a point R which divides the line joining P and Q in the ratio 2: 1 (i) internally and (ii) externally.
Solution:
i) The position vector of the point R dividing the joining of  P and Q internally in the ratio 2: 1 is
\(\overline{\mathrm{OR}}=\frac{2(\overline{\mathrm{a}}+\overline{\mathrm{b}})+(3 \overline{\mathrm{a}}-2 \overline{\mathrm{b}})}{2+1}=\frac{5 \bar{a}}{3}\)

ii) The position vector of the point R dividing the joining of P and Q externally in the ratio 2: 1 is
\(\overline{\mathrm{OR}}=\frac{2(\overline{\mathrm{a}}+\overline{\mathrm{b}})-(3 \overline{\mathrm{a}}-2 \overline{\mathrm{b}})}{2-1}=4 \overline{\mathrm{b}}-\overline{\mathrm{a}}\)

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 9.
Show that the points \(A(2 \bar{i}-\bar{j}+\bar{k}), \dot{B}(\overline{\mathbf{i}}-3 \overline{\mathbf{j}}-5 \overline{\mathbf{k}}), \mathbf{C}(3 \overline{\mathbf{i}}-4 \overline{\mathbf{j}}-4 \overline{\mathbf{k}})\) are the vertices of a right angled triangle.
Solution:
We have \(\overline{\mathrm{AB}}=\overline{\mathrm{OB}}-\overline{\mathrm{OA}}\)
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 9
∴ AB2 = BC2 + CA2 and hence a right angled triangle can be formed with the points A, B, and C.

Question 10.
Show that the points \(\mathbf{A}(2 \overline{\mathbf{i}}-\overline{\mathbf{j}}+\overline{\mathbf{k}}) \dot{B}(\bar{i}-3 \bar{j}-5 \bar{k}), C(3 \bar{i}-4 \bar{j}-4 \bar{k})\) are the vertices of a right angled triangle.
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 11

Question 11.
In a ΔABC if \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are the position vectors of the vertices A, B and C respectively, then prove that the position vector of the centroid G is \(\frac{1}{3}(\bar{a}+\bar{b}+\bar{c})\).
Solution:

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 13

Let G be the centroid of ΔABC and AD is the median through the vertex A.
Then AG : GD = 2: 1
Suppose \(\overline{\mathrm{OA}}=\overline{\mathrm{a}}, \overline{\mathrm{OB}}=\overline{\mathrm{b}}, \overline{\mathrm{OC}}=\overline{\mathrm{c}}\) with
reference to the specific origin O.
Mid point of BC is = \(\overline{\mathrm{OD}}=\frac{1}{2}(\overline{\mathrm{b}}+\overline{\mathrm{c}})\)
Since G divides AD in the ratio 2: 1 we have
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 14

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 12.
In a ΔABC, If ‘O’ Is the circumcentre, and H is the orthocentre then show that
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 15
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 16
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 17

Question 13.
Let \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}, \overline{\mathbf{d}}\) be the position vectors of A, B, C and D respectively which are the vertices of a tetrahedron. Then prove that the lines joining the vertices to the centroids of the opposite faces are concurrent. (this point is called the centroid or the centre of the tetrahedron)
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 18

Let O be the origin and G1, G2, G3, G4 be the centroids of ΔBCD, ΔCAD, ΔABD and ΔABC.
Then \(\overline{\mathrm{OG}}_1=\frac{\overline{\mathrm{b}}+\overline{\mathrm{c}}+\overline{\mathrm{d}}}{3}\)
Suppose P is the point which divides AG1 in the ratio 3: 1 then
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 19
Similarly position vectors of the points dividing BC2, CG3 and DG4 in the ratio 3: 1 are each equal to \(\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}+\overline{\mathrm{d}}}{4}\)
Hence the point P lies on AG1, BC2, CG3, DG4.

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 14.
Let OABC be a parallelogram and D is the midpoint of \(\overline{\mathbf{O A}}\). Prove that the segment is \(\overline{\mathbf{C D}}\) trisects the diagonal \(\overline{\mathbf{O B}}\) and is trisected by the diagonal\(\overline{\mathbf{O B}}\)
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 20
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 21

Question 15.
Let \(\overline{\mathbf{a}}, \overline{\mathbf{b}}\) be non-collinear vectors, if
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 22
are such that 3α = 2β then find x and y.
Solution:
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 23

Question 16.
If the points whose position vectors are \(3 \overline{\mathbf{i}}-2 \overline{\mathbf{j}}-\overline{\mathbf{k}}, \quad 2 \overline{\mathbf{i}}+3 \overline{\mathbf{j}}-4 \overline{\mathbf{k}},-\overline{\mathbf{i}}+\overline{\mathbf{j}}+2 \mathbf{k} 4 \overline{\mathbf{i}}+5 \overline{\mathbf{j}}+\lambda \overline{\mathbf{k}}\) are coplanar, then show that \(\lambda=\frac{-146}{17}\)
Solution:
Let O be the origin and let A, B, C and D be the given points. Then
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 25
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 26

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 17.
Find the equation of the line parallel to the vector \(2 \overline{\mathbf{i}}-\overline{\mathbf{j}}+2 \overline{\mathbf{k}}\) and which passes through the point A whose position vector is \(2 \bar{i}-\bar{j}+2 \bar{k}\). If P is a point on this line such that AP = 15, find the position vector of P.
Solution:
The vector equation of the line passing through the point \(\mathrm{A}(\bar{a})\) whose position vector is \(\overline{\mathrm{a}}=3 \overline{\mathrm{i}}+\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and parallel to the vector \(\overline{\mathrm{b}}=2 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}}\) is \(\overline{\mathrm{r}}=\overline{\mathrm{a}}+\mathrm{t} \overline{\mathrm{b}}\) for some t ∈ R
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 27

Question 18.
Show that the line joining the pair of points \(6 \bar{a}-4 \bar{b}+4 \bar{c},-4 \bar{c}\) and the line joining the pair of points \(-\bar{a}-2 \bar{b}-3 \bar{c}, \bar{a}+2 \bar{b}-5 \bar{c}\) intersect at the point \(-4 \bar{c}\) when \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) are non coplanar vectors.
Solution:
The vector equation of the line joining points
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 30
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 28
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 29
and equation (5) is satisfied.
∴ The two lines intersect at the point from (1) is \(-4 \bar{c}\)

TS Inter 1st Year Maths 1A Addition of Vectors Important Questions

Question 19.
Find the point of intersection of the line \(\overline{\mathbf{r}}=\mathbf{2} \overline{\mathbf{a}}+\overline{\mathbf{b}}+\mathbf{t}(\overline{\mathbf{b}}-\overline{\mathbf{c}})\) and the plane \(\overline{\mathbf{r}}=\overline{\mathbf{a}}+\mathbf{x}(\overline{\mathbf{b}}+\overline{\mathbf{c}})+\mathbf{y}(\overline{\mathbf{a}}+2 \overline{\mathbf{b}}-\overline{\mathbf{c}})\) where a, b, c are non-coplanar vectors.
Solution:
Let \(\overline{\mathbf{r}}\) be the position vector of the point P the intersection of the line and the plane.
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Question 20.
Prove that the vector equation of line through the points \(A(\bar{a}), B(\bar{b})\) is \(\overline{\mathbf{r}}=(\mathbf{1}-\mathbf{t}) \overline{\mathbf{a}}+\mathbf{t} \overline{\mathbf{b}}, \mathbf{t} \in \mathbf{R}\).
Solution:
Let O be the origin and P be any point on the line.
TS Inter 1st Year Maths 1A Addition of Vectors Important Questions 32

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