Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
and
Find and , when
and
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Find and , when
and
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Find and , when
and
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Find and , when
and
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Find λ if .
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒
If and , find .
Verify that (i) and are perpendicular to each other
and (ii) and are perpendicular to each other.
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
If and are perpendicular to each other then,
⇒
i.e.,
And in the similar way, we have,
Hence proved.
Find the value of:
i. ii. iii.
i.
The value of is, …
⇒
ii.
The value of is, … …
⇒
iii.
The value of is, … …
⇒
Find the unit vectors perpendicular to both and when
and
Let be the vector which is perpendicular to then we have,
…where k is a scalor
Thus, we have r is a unit vector,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒
Find the unit vectors perpendicular to both and when
and
Let be the vector which is perpendicular to then we have,
…where k is a scalar
Thus, we have r is a unit vector,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒
Find the unit vectors perpendicular to both and when
and
Let be the vector which is perpendicular to then we have,
…where k is a scalar
Thus, we have r is a unit vector,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒
Find the unit vectors perpendicular to both and when
and
Let be the vector which is perpendicular to then we have,
…where k is a scalar
Thus, we have r is a unit vector,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒
Find the unit vectors perpendicular to the plane of the vectors
and
Let be the vector which is perpendicular to then we have,
…where k is a scalar
Thus, we have r is a unit vector,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Find a vector of magnitude 6 which is perpendicular to both the vectors
and .
Let be the vector which is perpendicular to then we have,
…where k is a scalar
Thus, we have r is vector of magnitude 6,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Here, as r is of magnitude 6 thus,
k = 6,
Thus,
Find a vector of magnitude 5 units, perpendicular to each of the vectors
and, where and
Let be the vector which is perpendicular to then we have,
…where k is a scalar
Thus, we have r is vector of magnitude 5,
So,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Here, as r is of magnitude 5 thus,
k = 5,
Thus,
Find an angle between two vectors and with magnitudes 1 and 2 respectively and .
We are given that and .
And,
So we have,
|| =
⇒
⇒
⇒
If , and , find a vector which is perpendicular to both and and for which .
Given that
Let be the vector which is perpendicular to then we have,
…where k is a scalar
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Given that
⇒
⇒
⇒
If , and , find a vector which is perpendicular to both and and for which .
Given that
Let be the vector which is perpendicular to then we have,
…where k is a scalar
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
Given that
⇒
⇒
Prove that , where θ is the angle between and .
We know that |
And |
So,
Hence, proved.
Write the value of p for which and are parallel vectors.
As the vectors are parallel vectors so,
Thus,
We have,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ Thus, p = .
Verify that , when
, and
To verify
We need to prove L.H.S = R.H.S
L.H.S we have,
Given,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
RHS is
⇒
Thus, LHS = RHS.
Verify that , when
, and .
To verify
We need to prove L.H.S = R.H.S
L.H.S we have,
Given, ,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
RHS is
⇒
Thus, LHS = RHS.
Find the area of the parallelogram whose adjacent sides are represented by the vectors:
and
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Find the area of the parallelogram whose adjacent sides are represented by the vectors:
and
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Find the area of the parallelogram whose adjacent sides are represented by the vectors:
and
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Find the area of the parallelogram whose adjacent sides are represented by the vectors:
and
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Find the area of the parallelogram whose diagonal are represented by the vectors
and
The diagonals are
Thus,
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒ sq units
Find the area of the parallelogram whose diagonal are represented by the vectors
and
The diagonals are
Thus,
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
,
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒ sq units
Find the area of the parallelogram whose diagonal are represented by the vectors
and .
The diagonals are
Thus,
The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
⇒ sq units
Find the area of the triangle whose two adjacent sides are determined by the vectors
and
The area of the triangle = , where a and b are it’s adjacent sides vectors.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Find the area of the triangle whose two adjacent sides are determined by the vectors
and .
The area of the triangle = , where a and b are it’s adjacent sides vectors.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Using vectors, find the area of ΔABC whose vertices are
A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)
Through the vertices we get the adjacent vectors as,
The area of the triangle = , where a and b are it’s adjacent sides vectors.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Using vectors, find the area of ΔABC whose vertices are
A(1, 2, 3), B(2, −1, 4) and C(4, 5, Δ1) ((considering Δ1 as 1 ))
Through the vertices we get the adjacent vectors as,
The area of the triangle = , where a and b are it’s adjacent sides vectors.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Using vectors, find the area of ΔABC whose vertices are
A(3, −1, 2), B(1, −1, −3) and C(4, −3, 1)
Through the vertices we get the adjacent vectors as,
The area of the triangle = , where a and b are it’s adjacent sides vectors.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Using vectors, find the area of ΔABC whose vertices are
A(1, −1, 2), B(2, 1, −1) and C(3, −1, 2).
Through the vertices we get the adjacent vectors as,
The area of the triangle = , where a and b are it’s adjacent sides vectors.
Area =
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒ sq units
Using vector method, show that the given points A, B, C are collinear:
A(3, −5, 1), B(−1, 0, 8) and C(7, −10, −6)
Through the vertices we get the adjacent vectors as,
To prove that A, B, C are collinear we need to prove that
.
So,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
Using vector method, show that the given points A, B, C are collinear:
A(6, −7, −1), B(2, −3, 1) and C(4, −5, 0).
Through the vertices we get the adjacent vectors as,
To prove that A, B, C are collinear we need to prove that
.
So,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
Thus, A, B and C are collinear.
Show that the point A, B, C with position vectors , and respectively are collinear.
Through the vertices we get the adjacent vectors as,
To prove that A, B, C are collinear we need to prove that
.
So,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
Thus, A, B and C are collinear.
Show that the points having position vectors are collinear, whatever be .
Through the vertices we get the adjacent vectors as,
To prove that A, B, C are collinear we need to prove that
.
So,
Here,
We
have
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
Thus, A, B and C are collinear.
Show that the points having position vector , and are collinear, whatever be .
We have,
Through the vertices we get the adjacent vectors as,
To prove that A, B, C are collinear we need to prove that
.
So,
Here,
We
have
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
Thus, A, B and C are collinear.
Find a unit vector perpendicular to the plane ABC, where the points A, B, C, are , and respectively.
A unit vector perpendicular to the plane ABC will be,
Through the vertices we get the adjacent vectors as,
Here,
We
have
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
⇒
If and then find .
and
Then,
We have,
Here,
We
have
and
⇒
Thus, substituting the values of ,
in equation (i) we get
⇒
⇒
If , and , find .
We have,
So,
⇒
⇒
If , and , find the angle between and .
We have,
⇒
⇒
⇒
⇒
⇒
⇒