Page No 24.29:
Question 1:
Find when
(i)
(ii)
(iii)
Answer:
Page No 24.30:
Question 2:
For what value of λ are the vectors perpendicular to each other if
(i)
(ii)
(iii)
(iv)
Answer:
Page No 24.30:
Question 3:
If are two vectors such that , find the angle between
Answer:
Page No 24.30:
Question 4:
Answer:
Page No 24.30:
Question 5:
Find the angle between the vectors , where
(i)
(ii)
(iii)
(iv)
(v)
Answer:
Page No 24.30:
Question 6:
Find the angles which the vector makes with the coordinate axes.
Answer:
Page No 24.30:
Question 7:
(i) Dot product of a vector with are 0, 5 and 8 respectively. Find the vector.
(ii) Dot products of a vector with vectors are respectively 4, 0 and 2. Find the vector.
Answer:
Page No 24.30:
Question 8:
If are unit vectors inclined at an angle θ, prove that
(i)
(ii)
Answer:
Page No 24.30:
Question 9:
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is .
Answer:
Page No 24.30:
Question 10:
If are three mutually perpendicular unit vectors, then prove that .
Answer:
Page No 24.30:
Question 11:
If
Answer:
Page No 24.30:
Question 12:
Show that the vector is equally inclined to the coordinate axes.
Answer:
Page No 24.30:
Question 13:
Show that the vectors are mutually perpendicular unit vectors.
Answer:
Page No 24.30:
Question 14:
For any two vectors , show that .
Answer:
Page No 24.30:
Question 15:
If , and , find λ such that is perpendicular to . [NCERT EXEMPLAR]
Answer:
The given vectors are , and .
Now,
It is given that
Thus, the value of λ is −2.
Page No 24.30:
Question 16:
If then find the value of λ, so that and are perpendicular vectors.
Answer:
Page No 24.30:
Question 17:
If then express in the form of where is parallel to is perpendicular to .
Answer:
Page No 24.31:
Question 18:
If either , then But the converse need not be true. Justify your answer with an example.
Answer:
Page No 24.31:
Question 19:
Show that the vectors form a right-angled triangle.
Answer:
Page No 24.31:
Question 20:
If are such that is perpendicular to , then find the value of λ.
Answer:
Page No 24.31:
Question 21:
Find the angles of a triangle whose vertices are A (0, −1, −2), B (3, 1, 4) and C (5, 7, 1).
Answer:
Page No 24.31:
Question 22:
Find the magnitude of two vectors that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.
Answer:
Page No 24.31:
Question 23:
Show that the points whose position vectors are form a right triangle.
Answer:
Page No 24.31:
Question 24:
If the vertices A, B and C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC?
Answer:
Page No 24.31:
Question 25:
If A, B and C have position vectors (0, 1, 1), (3, 1, 5) and (0, 3, 3) respectively, show that ∆ ABC is right-angled at C.
Answer:
Page No 24.31:
Question 26:
Find the projection of , where
Answer:
Page No 24.31:
Question 27:
If then show that the vectors are orthogonal.
Answer:
Page No 24.31:
Question 28:
A unit vector makes angles with and respectively and an acute angle θ with . Find the angle θ and components of .
Answer:
Page No 24.31:
Question 29:
If two vectors are such that then find the value of
Answer:
Page No 24.31:
Question 30:
If is a unit vector, then find in each of the following.
(i)
(ii)
Answer:
Page No 24.31:
Question 31:
Find if
(i)
(ii)
(iii)
Answer:
Page No 24.31:
Question 32:
Find if
(i)
(ii)
(iii)
Answer:
Page No 24.31:
Question 33:
Find the angle between two vectors if
(i)
(ii)
Answer:
Page No 24.32:
Question 34:
Express the vector as the sum of two vectors such that one is parallel to the vector and other is perpendicular to .
Answer:
Page No 24.32:
Question 35:
If are two vectors of the same magnitude inclined at an angle of 30°, such that
Answer:
Page No 24.32:
Question 36:
Express as the sum of a vector parallel and a vector perpendicular to
Answer:
Page No 24.32:
Question 37:
Decompose the vector into vectors which are parallel and perpendicular to the vector
Answer:
Page No 24.32:
Question 38:
Let Find λ such that is orthogonal to .
Answer:
Page No 24.32:
Question 39:
If what can you conclude about the vector ?
Answer:
Page No 24.32:
Question 40:
If is perpendicular to both , then prove that it is perpendicular to both .
Answer:
Page No 24.32:
Question 41:
If prove that
Answer:
Page No 24.32:
Question 42:
If are three non-coplanar vectors, such that then show that is the null vector.
Answer:
Given that:
so, either = 0 or
similarly,
so, = 0 or
Also,
so, = 0 or
But cannot be perpendicular to as are non-coplanar.
so, =0. is a null vector.
Page No 24.32:
Question 43:
If a vector is perpendicular to two non-collinear vectors is perpendicular to every vector in the plane of
Answer:
Page No 24.32:
Question 44:
If show that the angle θ between the vectors is given by
cos θ =
Answer:
Page No 24.32:
Question 45:
Let be vectors such If then find
Answer:
Page No 24.32:
Question 46:
Let be three vectors. Find the values of x for which the angle between is acute and the angle between is obtuse.
Answer:
Page No 24.32:
Question 47:
Find the values of x and y if the vectors are mutually perpendicular vectors of equal magnitude.
Answer:
Page No 24.32:
Question 48:
If are two non-collinear unit vectors such that find
Answer:
Page No 24.33:
Question 49:
If , are two vectors such that , then prove that is perpendicular to .
Answer:
Page No 24.46:
Question 1:
In a triangle OAB, AOB = 90º. If P and Q are points of trisection of AB, prove that .
Answer:
In triangle OAB, AOB = 90º. P and Q are points of trisection of AB.
Taking O as the origin, let the position vectors of A and B be and , respectively.
Since P and Q are the points of trisection of AB, so AP : PB = 1 : 2 and AQ : QB = 2 : 1.
Position vector of P, (Using section formula)
Position vector of Q,
Also, .
.....(1)
Now,
Page No 24.46:
Question 2:
Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Answer:
Let OACB be a quadrilateral such that diagonals OC and AB bisect each other at 90º.
Taking O as the origin, let the poisition vectors of A and B be and , respectively. Then,
and
Position vector of mid-point of AB,
∴ Position vector of C,
By the triangle law of vector addition, we have
Since ,
In a quadrilateral if diagonals bisects each other at right angle and adjacent sides are equal, then it is a rhombus.
Page No 24.46:
Question 3:
(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Answer:
Let ABC be a right triangle with BAC = 90º. Taking A as the origin, let the position vectors of B and C be and , respectively. Then,
and
Since .....(1)
Now,
.....(2)
Also,
From (2) and (3), we have
Page No 24.46:
Question 4:
Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
Answer:
Let ABCD be a parallelogram such that AC and BD are its two diagonals. Taking A as the origin, let the position vectors of B and D be and , respectively. Then,
and
Using triangle law of vector addition, we have
In ∆ABC,
Now,
Also,
From (1) and (2), we have
Page No 24.46:
Question 5:
Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
Answer:
ABCD is a rectangle. Let P, Q, R and S be the mid-points of the sides AB, BC, CD and DA, respectively.
Now,
.....(1)
.....(2)
From (1) and (2), we have
So, the sides PQ and SR are equal and parallel. Thus, PQRS is a parallelogram.
Now,
Also,
From (3) and (4), we have
So, the adjacent sides of the parallelogram are equal. Hence, PQRS is a rhombus.
Page No 24.46:
Question 6:
Prove that the diagonals of a rhombus are perpendicular bisectors of each other.
Answer:
Let OABC be a rhombus, whose diagonals OB and AC intersect at D. Suppose O is the origin.
Let the position vector of A and C be and , respectively. Then,
and
In ∆OAB,
Position vector of mid-point of
Position vector of mid-point of (Mid-point formula)
So, the mid-points of OB and AC coincide. Thus, the diagonals OB and AC bisect each other.
Now,
Hence, the diagonals OB and AC are perpendicular to each other.
Thus, the diagonals of a rhombus are perpendicular bisectors of each other.
Page No 24.46:
Question 7:
Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.
Answer:
Let ABCD be a rectangle. Take A as the origin.
Suppose the position vectors of points B and D be and , respectively.
Now,
Also,
Since ABCD is rectangle, so .
.....(1)
Now, diagonals AC and BD are perpendicular
iff
iff ABCD is a square
Thus, the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.
Page No 24.46:
Question 8:
If AD is the median of ∆ABC, using vectors, prove that .
Answer:
Taking A as the origin, let the position vectors of B and C be and , respectively.
It is given that AD is the median of ∆ABC.
∴ Position vector of mid-point of BC = (Mid-point formula)
Now,
.....(1)
Also,
From (1) and (2), we have
Page No 24.46:
Question 9:
If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.
Answer:
Let ∆ABC be a triangle such that AD is the median. Taking A as the origin, let the position vectors of B and C be and , respectively. Then,
Position vector of D = (Mid-point formula)
Now,
= Position vector of D − Position vector of A =
= Position vector of C − Position vector of B =
Since ,
Hence, the ∆ABC is an isosceles triangle.
Page No 24.46:
Question 10:
In a quadrilateral ABCD, prove that , where P and Q are middle points of diagonals AC and BD.
Answer:
Let ABCD be the quadrilateral. Taking A as the origin, let the position vectors of B, C and D be and , respectively. Then,
Position vector of P = (Mid-point formula)
Position vector of Q = (Mid-point formula)
Now,
Also,
From (1) and (2), we have
Page No 24.46:
Question 1:
What is the angle between vectors with magnitudes 2 and respectively? Given
Answer:
Page No 24.46:
Question 2:
are two vectors such that Write the projection of .
Answer:
Page No 24.46:
Question 3:
Find the cosine of the angle between the vectors
Answer:
Page No 24.46:
Question 4:
If the vectors are orthogonal, find m.
Answer:
Page No 24.46:
Question 5:
If the vectors are parallel, find the value of m.
Answer:
Page No 24.46:
Question 6:
If are vectors of equal magnitude, write the value of
Answer:
Page No 24.47:
Question 7:
If are two vectors such that find the relation between the magnitudes of .
Answer:
Page No 24.47:
Question 8:
For any two vectors , write when holds.
Answer:
Page No 24.47:
Question 9:
For any two vectors , write when holds.
Answer:
Page No 24.47:
Question 10:
If are two vectors of the same magnitude inclined at an angle of 60° such that write the value of their magnitude.
Answer:
Page No 24.47:
Question 11:
If what can you conclude about the vector ?
Answer:
Page No 24.47:
Question 12:
If is a unit vector such that
Answer:
Page No 24.47:
Question 13:
If are unit vectors such that is a unit vector, write the value of
Answer:
Page No 24.47:
Question 14:
If
Answer:
Page No 24.47:
Question 15:
If find the projection of .
Answer:
Page No 24.47:
Question 16:
For any two non-zero vectors, write the value of
Answer:
Page No 24.47:
Question 17:
Write the projections of on the coordinate axes.
Answer:
Page No 24.47:
Question 18:
Write the component of along .
Answer:
Page No 24.47:
Question 19:
Write the value of where is any vector.
Answer:
Page No 24.47:
Question 20:
Find the value of θ ∈(0, π/2) for which vectors are perpendicular.
Answer:
Page No 24.47:
Question 21:
Write the projection of along the vector .
Answer:
Page No 24.47:
Question 22:
Write a vector satisfying
Answer:
Page No 24.47:
Question 23:
If are unit vectors, find the angle between
Answer:
Page No 24.47:
Question 24:
If are mutually perpendicular unit vectors, write the value of
Answer:
Page No 24.47:
Question 25:
If are mutually perpendicular unit vectors, write the value of
Answer:
Page No 24.47:
Question 26:
Find the angle between the vectors
Answer:
Page No 24.47:
Question 27:
For what value of λ are the vectors perpendicular to each other?
Answer:
Page No 24.47:
Question 28:
Find the projection of
Answer:
Page No 24.47:
Question 29:
Write the value of p for which are parallel vectors.
Answer:
Page No 24.47:
Question 30:
Find the value of λ if the vectors are perpendicular to each other.
Answer:
Page No 24.48:
Question 31:
If find the projection of .
Answer:
Page No 24.48:
Question 32:
Write the angle between two vectors with magnitudes and 2 respectively if
Answer:
Page No 24.48:
Question 33:
Write the projection of the vector on the vector .
Answer:
We know that projection of on = .
Let and .
∴ Projection of on
Page No 24.48:
Question 34:
Find λ when the projection of is 4 units.
Answer:
Page No 24.48:
Question 35:
For what value of λ are the vectors perpendicular to each other?
Answer:
Page No 24.48:
Question 36:
Write the projection of the vector on the vector
Answer:
Page No 24.48:
Question 37:
Write the value of λ so that the vectors are perpendicular to each other.
Answer:
Page No 24.48:
Question 38:
Write the projection of
Answer:
Page No 24.48:
Question 39:
If and are perpendicular vectors, and , find the value of . [CBSE 2014]
Answer:
Disclaimer: has been taken in order to solve the question.
It is given that and are perpendicular vectors.
.....(1)
Thus, the value of is 12.
Page No 24.48:
Question 40:
If the vectors and are such that and is a unit vector, then write the angle between and . [CBSE 2014]
Answer:
Let the angle between and be .
It is given that is a unit vector.
Thus, the angle between and is .
Page No 24.48:
Question 41:
If and are two unit vectors such that is also a unit vector, then find the angle between and . [CBSE 2014]
Answer:
Let the angle between and be .
It is given that .
Thus, the angle between and is .
Page No 24.48:
Question 42:
If and are unit vectors, then find the angle between and , given that is a unit vector. [CBSE 2014]
Answer:
Let the angle between and be .
It is given that .
Thus, the angle between and is .
Page No 24.48:
Question 43:
Find the magnitude of each of the two vectors , having the same magnitude such that the angle between them is 60° and their scalar product is
Answer:
Let the two vectors be .
Since the vectors have same magnitude so,
Scalar product of two vectors =
Page No 24.49:
Question 1:
The vectors satisfy the equations If θ is the angle between , then
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 24.49:
Question 2:
If then
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 24.49:
Question 3:
If then the angle between is
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 24.49:
Question 4:
Let be two unit vectors and α be the angle between them. Then, is a unit vector if
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 24.49:
Question 5:
The vector (cos α cos β) + (cos α sin β) + (sin α) is a
(a) null vector
(b) unit vector
(c) constant vector
(d) None of these
Answer:
(b) unit vector
Page No 24.49:
Question 6:
If the position vectors of P and Q are then the cosine of the angle between and y-axis is
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 24.49:
Question 7:
If are unit vectors, then which of the following values of is not possible?
(a)
(b)
(c)
(d) −1/2
Answer:
(a)
Page No 24.49:
Question 8:
If the vectors are perpendicular, then the locus of (x, y) is
(a) a circle
(b) an ellipse
(c) a hyperbola
(d) None of these
Answer:
(b) an ellipse
Page No 24.49:
Question 9:
The vector component of perpendicular to is
(a)
(b)
(c)
(d) None of these
Answer:
(b)
Page No 24.49:
Question 10:
What is the length of the longer diagonal of the parallelogram constructed on if it is given that and the angle between is π/4?
(a) 15
(b)
(c)
(d)
Answer:
(c)
Page No 24.50:
Question 11:
If is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ is a unit vector if
(a) λ = 1
(b) λ = −1
(c) a = |λ|
(d)
Answer:
(d)
Page No 24.50:
Question 12:
If θ is the angle between two vectors only when
(a)
(b)
(c) 0 < θ < π
(d) 0 ≤ θ ≤ π
Answer:
(b)
Page No 24.50:
Question 13:
The values of x for which the angle between is obtuse and the angle between and the z-axis is acute and less than are
(a)
(b)
(c)
(d) ϕ
Answer:
(b)
Let the angle between vector a and vector b be A.
Page No 24.50:
Question 14:
If are any three mutually perpendicular vectors of equal magnitude a, then is equal to
(a) a
(b)
(c)
(d) 2a
(e) None of these
Answer:
(c)
Page No 24.50:
Question 15:
If the vectors are perpendicular, then λ is equal to
(a) −14
(b) 7
(c) 14
(d)
Answer:
(c) 14
Page No 24.50:
Question 16:
The projection of the vector along the vector of is
(a) 1
(b) 0
(c) 2
(d) −1
(e) −2
Answer:
(a) 1
Page No 24.50:
Question 17:
The vectors and are perpendicular if
(a) a = 2, b = 3, c = −4
(b) a = 4, b = 4, c = 5
(c) a = 4, b = 4, c = −5
(d) a = −4, b = 4, c = −5
Answer:
(b) a = 4, b = 4, c = 5
Page No 24.50:
Question 18:
If
(a) positive
(b) negative
(c) 0
(d) None of these
Answer:
(c) 0
Page No 24.50:
Question 19:
If are unit vectors inclined at an angle θ, then the value of is
(a)
(b) 2 sin θ
(c)
(d) 2 cos θ
Answer:
(a)
Page No 24.50:
Question 20:
If are unit vectors, then the greatest value of is
(a) 2
(b)
(c) 4
(d) None of these
Answer:
(c) 4
Page No 24.50:
Question 21:
If the angle between the vectors is acute, then x lies in the interval
(a) (−4, 7)
(b) [−4, 7]
(c) R −[−4, 7]
(d) R −(4, 7)
Answer:
(c) R −[−4, 7]
Page No 24.50:
Question 22:
If are two unit vectors inclined at an angle θ, such that then
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 24.51:
Question 23:
Let be three unit vectors, such that =1 and is perpendicular to . If makes angles α and β with respectively, then cos α + cos β =
(a)
(b)
(c) 1
(d) −1
Answer:
(d) −1
Page No 24.51:
Question 24:
The orthogonal projection of is
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 24.51:
Question 25:
If θ is an acute angle and the vector (sin θ) + (cos θ) is perpendicular to the vector then θ =
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 24.51:
Question 26:
If be two unit vectors and θ the angle between them, then is a unit vector if θ =
(a)
(b)
(c)
(d)
Answer:
(d)
View NCERT Solutions for all chapters of Class 15