**7 ** Show that the vectors coplanar if are coplanar

vectors will be coplanar if
now, it is given that
now as we know that whenever two vectors are same in a vector triple product its value becomes zero, so
Hence when is zero

**6 ** Find x such that the four points A (3, 2, 1) B (4, x, 5), C (4, 2, –2) and D (6, 5, –1) are coplanar.

Given
four points A = (3, 2, 1) , B= (4, x, 5), C= (4, 2, –2) and D =(6, 5, –1) .
These four points will be coplanar when the volume of the shape they make will be zero. which means
Hence the value of x is 5.

**5** Show that the four points with position vectors and are coplanar.

Given four position vectors,
And,
now these will be coplanar when
Now, let's calculate the vector triple product of these vectors
Hence, the four points are coplanar.

**4(b)** Let Then

If and show that no value of can make coplanar.

Given
these will be coplanar when,
Hence the value of is -2, irrespective of the value of . Hence no value of satisfies the condition of three vectors being coplanar.

**4 (a) ** Let Then

If and find which makes coplanar

Given
These will be coplanar when
Hence the value of for which vectors will be coplanar is 2.

**3. **Find if the vectors are coplanar.

**2. **Show that the vectors are coplanar.

Given
For coplanarity,
So let's calculate the vector triple product of these three vectors,
Hence the three vectors are coplanar.

**1** Find

**Q19 ** Choose the correct. If is the angle between any two vectors , then when

is equal to

Given in the question
is the angle between any two vectors and
To find the value of
Hence option D is correct.

**Q18 ** The value of is

(A) 0

(B) –1

(C) 1

(D) 3

**Q17** Choose the correct answer. Let be two unit vectors and is the angle between them. Then is a unit vector if

Gicen in the question
be two unit vectors and is the angle between them
also
Then is a unit vector if
Hence option D is correct.

**Q.14 **If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to .

Given
and
Now, let vector is inclined to at respectively.
Now,
Now, Since,
Hence vector is equally inclined to .

**Q16 **Choose the correct answer If is the angle between two vectors , then only when

Given in the question
is the angle between two vectors
this will satisfy when
Hence option B is the correct answer.

**Q15** Prove that , if and only if are perpendicular, given

.

Given in the question,
are perpendicular and we need to prove that
LHS=
if are perpendicular,
= RHS
LHS ie equal to RHS
Hence proved.

**Q13 ** The scalar product of the vector with a unit vector along the sum of vectors and is equal to one. Find the value of .

Let, the sum of vectors and be
unit vector along
Now, the scalar product of this with
squaring both the side,

**Q12 **Let . Find a vector which is perpendicular to both

Given,
Let
now, since it is given that d is perpendicular to and , we got the condition,
and
And
And
here we got 2 equation and 3 variable. one more equation will come from the condition:
so now we have three equation and three variable,
On solving this three equation we get,
,
Hence Required vector :
.

**Q11** Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are

Let a vector is equally inclined to axis OX, OY and OZ.
let direction cosines of this vector be
Now
Hence direction cosines are:
.

**Q10 ** The two adjacent sides of a parallelogram are . Find the unit vector parallel to its diagonal. Also, find its area.

Given, two adjacent sides of the parallelogram
The diagonal will be the resultant of these two vectors. so
resultant R:
Now unit vector in direction of R
Hence unit vector along the diagonal of the parallelogram
Now,
Area of parallelogram
Hence the area of the parallelogram is .

**Q9 ** Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Given, two vectors
the point R which divides line segment PQ in ratio 1:2 is given by
Hence position vector of R is .
Now, Position vector of the midpoint of RQ
which is the position vector of Point P . Hence, P is the mid-point of RQ

**Q8** Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

Given in the question,
points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)
now let's calculate the magnitude of the vectors
As we see that AB = BC + AC, we conclude that three points are colinear.
we can also see from here,
Point B divides AC in the ratio 2 : 3.

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