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Show that the vectors $\vec a , \vec b , \vec c$  coplanar if $\vec a + \vec b , \vec b + \vec c , \vec c + \vec a$ 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

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 $4\hat i + 8 \hat j +12\hat k, 2\hat i+ 4\hat j + 6\hat k,3\hat i+ 5\hat j + 4 \hat k$ and $5\hat i + 8 \hat j +5 \hat k$  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 $\vec a = \hat i + \hat j + \hat k , \vec b = \hat i \: \: and \: \: \vec c = c_1 \hat i + c_2 \hat j + c_3 \hat k$  Then
If $c_2 =- 1$ and $c_3 =1$ show that no value of $c_3$ can make $\vec a , \vec b , \vec c$ 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 $\vec a = \hat i + \hat j + \hat k , \vec b = \hat i \: \: and \: \: \vec c = c_1 \hat i + c_2 \hat j + c_3 \hat k$  Then
If $c_1 = 1$ and $c_2 = 2$  find $c_3$ which makes $\vec a , \vec b , \vec c$ coplanar

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

3. Find $\lamda$$\lambda$ if the vectors $\hat i - \hat j +\hat k ,3 \hat i+\hat j +\hat 2k\: \: and\: \: \hat i+ \lambda \hat j - 3 \hat k$ are coplanar.

Given three vectors, These three will be coplanar when,

2. Show that the vectors $\vec a = \hat i - 2 \hat j + 3 \hat k, \vec b =- 2 \hat i + 3 \hat j -4\hat k \: \:and \: \: \vec c = \hat i -3 \hat j +5 \hat k$  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 $[ \vec a\: \: \vec b \: \: \vec c ]\: \: i\! f \: \: \vec a = \hat i - 2 \hat j + 3 \hat k, \vec b = 2 \hat i - 3 \hat j + \hat k \: \:and \: \: \vec c = 3 \hat i + \hat j - 2 \hat k$

Given, Hence the value of  is 24.

Q19  Choose the correct. If  $\theta$ is the angle between any two vectors  $\vec a \: \:and \: \: \vec b$ , then  $|\vec a \cdot \vec b |=|\vec a \times \vec b |$ when $\theta$
is equal to

$\\A ) 0 \\\\ B ) \pi /4 \\\\ C ) \pi / 2 \\\\ D ) \pi$

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  $\hat i ( \hat j \times \hat k ) + \hat j ( \hat i \times \hat k ) + \hat k ( \hat i \times \hat j )$ is

(A) 0

(B) –1

(C) 1

(D) 3

To find the value of   Hence option C is correct.

Q17  Choose the correct answer. Let $\vec a \: \: and \: \: \vec b$  be two unit vectors and $\theta$  is the angle between them. Then $\vec a + \vec b$is a unit vector if

$\\A ) \theta = \frac{\pi }{4} \\\\ B ) \theta = \frac{\pi }{3} \\\\ C ) \theta = \frac{\pi }{2} \\\\ D ) \theta = \frac{2\pi }{3}$

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 $\vec a , \vec b , \vec c$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec a+\vec b +\vec c$  is equally inclined to $\vec a , \vec b \: \: and \: \: \vec c$ .

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 $\theta$ is the angle between two vectors $\vec a \: \: and \: \: \vec b$  , then $\vec a \cdot \vec b \geq 0$ only when
$\\A ) 0 < \theta < \frac{\pi }{2} \\\\ \: \: \: \: B ) 0 \leq \theta \leq \frac{\pi }{2} \\\\ \: \: \: C ) 0 < \theta < \pi \\\\ \: \: \: D) 0 \leq \theta \leq\pi$

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

Q15  Prove that $( \vec a + \vec b ) . (\vec a + \vec b ) = |\vec a ^2 | + |\vec b |^2$ , if and only if  $\vec a , \vec b$ are perpendicular, given $\vec a \neq 0 , \vec b \neq 0$

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 $\hat i + \hat j + \hat k$ with a unit vector along the sum of vectors$2\hat i + 4 \hat j -5 \hat k$  and $\lambda \hat i + 2 \hat j +3 \hat k$ 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 $\vec a = \hat i + 4 \hat j + 2 \hat k , \vec b = 3 \hat i - 2 \hat j + 7 \hat k \: \:and \: \: \vec c = 2 \hat i - \hat j + 4 \hat k$  . Find a vector $\vec d$ which is perpendicular to both  $\vec a \: \: and \: \: \vec b \: \: and \: \: \vec c . \vec d = 15$

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 $\pm \left ( \frac{1}{\sqrt 3 } , \frac{1}{\sqrt 3 } , \frac{1}{\sqrt 3 } \right )$

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 $2 \hat i - 4 \hat j + 5 \hat k \: \:and \: \: \hat i - 2 \hat j - 3 \hat k$ . 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 $( 2 \vec a + \vec b ) \: \:and \: \: ( \vec a - 3 \vec b )$ 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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