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Q1. Show that the vectors

and

are coplanar

Solution

The vectors are coplanar, if = 0 _{$a with rightwards arrow on top equals i minus 2 j plus 3 k comma space space b with rightwards arrow on top equals negative 2 i plus 3 j minus 4 k comma space space c with rightwards arrow on top equals i minus 3 j plus 5 k b with rightwards arrow on top x c with rightwards arrow on top equals open vertical bar table row i j k row cell negative 2 end cell 3 cell negative 4 end cell row 1 cell negative 3 end cell 5 end table close vertical bar equals i left parenthesis 15 minus 12 right parenthesis minus j left parenthesis negative 10 plus 4 right parenthesis plus k left parenthesis 6 minus 3 right parenthesis equals 3 i plus 6 j plus 3 k a with rightwards arrow on top. left parenthesis b with rightwards arrow on top x c with rightwards arrow on top right parenthesis equals left parenthesis i minus 2 j plus 3 k right parenthesis. left parenthesis 3 i plus 6 j plus 3 k right parenthesis equals 3 minus 12 plus 9 equals 0 S o space a with rightwards arrow on top comma b with rightwards arrow on top comma c with rightwards arrow on top space a r e space c o p l a n a r$}

Q2.

Solution

Q3. Find the scalar product of the vectors

and

Solution

Let and

Q4. Find the scalar projection of the vector

on the vector

Solution

Scalar projection on

Q5. Write the direction ratio’s of the vector

and hence calculate its direction cosines.

Solution

The direction ratios of the vector are 1, 2, – 3 and

t h e space d i r e c t i o n space cos i n e s space a r e space fraction numerator 1 over denominator square root of 1 plus 4 plus 9 end root end fraction comma fraction numerator 2 over denominator square root of 1 plus 4 plus 9 end root end fraction comma fraction numerator negative 3 over denominator square root of 1 plus 4 plus 9 end root end fraction rightwards double arrow fraction numerator 1 over denominator square root of 14 end fraction comma fraction numerator 2 over denominator square root of 14 end fraction comma fraction numerator negative 3 over denominator square root of 14 end fraction

Q6.

Solution

Q7. Prove sine formula using vectors,

Solution

Let ABC be a triangle, such that are the vectors opposite to the angles respectively.

Similarly, we can find By using equations (1) and (2), we get

Q8. If

are unit vector such that

then find the value of

Solution

Here, we have

Q9.

Solution

L e t space a with rightwards arrow on top equals 3 i plus j minus 2 k space space b with rightwards arrow on top equals i minus 3 j minus 4 k t h e n space a with rightwards arrow on top x b with rightwards arrow on top equals open vertical bar table row i j k row 3 1 cell negative 2 end cell row 1 cell negative 3 end cell cell negative 4 end cell end table close vertical bar equals i left parenthesis negative 4 minus 6 right parenthesis minus j left parenthesis negative 12 plus 2 right parenthesis plus k left parenthesis negative 9 minus 1 right parenthesis equals negative 10 i plus 10 j minus 10 k a r e a space o f space p a r a l l e log r a m equals space vertical line a with rightwards arrow on top x b with rightwards arrow on top vertical line equals square root of 100 plus 100 plus 100 end root equals 10 square root of 3 s q space u n i t s

Q10. If

and

. Find the angle between

Solution

Here, we have Squaring on both sides, we get Let be the angle between the vectors. Thus, the angle between is 60°.

Q11. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are

respectively, in the ratio 2 : 3 internally.

Solution

Let m=2 and n= 3

Q12. If

and

represent two sides of a parallelogram, find unit vector parallel to the diagonals of the parallelogram.

Solution

Let OPQR is a parallelogram such that and . {

rightwards arrow for O R equals of rightwards arrow for P Q of

}

Q13. Find the value of x & y so that the vectors

are equal.

Solution

The vectors are equal, if and only if there corresponding components are equal ie x = 6 and y = 2x i.e., x = 6 and y = 12.

Q14. Find the angle between the vectors

and

Solution

Let be the angle between the vectors and.

Q15. If

, then show that

Solution

Here, we have So, by the equations (1) and (2), we get

Q16.

Solution

Q17. Find the unit vector in the direction of

begin mathsize 11px style AB with rightwards arrow on top end style

, where A and B are the points (2, – 3, 7) and (1, 3, – 4).

Solution

begin mathsize 11px style The space position space vector space of space point space straight A space is space begin space mathsize space OA with rightwards arrow on top space equals 2 straight i with hat on top minus 3 straight j with hat on top plus 7 straight k with hat on top space space and space OB with rightwards arrow on top space equals straight i with hat on top plus 3 straight j with hat on top minus 4 straight k with hat on top comma Then space AB with rightwards arrow on top space equals stack space OB with rightwards arrow on top space minus stack space OA with rightwards arrow on top space equals open parentheses straight i with hat on top plus 3 straight j with hat on top minus 4 straight k with hat on top close parentheses minus open parentheses 2 straight i with hat on top minus 3 straight j with hat on top plus 7 straight k with hat on top close parentheses rightwards double arrow AB with rightwards arrow on top space equals negative straight i with hat on top plus 6 straight j with hat on top minus 11 straight k with hat on top Now comma space unit space vector space of space AB with rightwards arrow on top space is AB with rightwards arrow on top equals fraction numerator negative straight i with hat on top plus 6 straight j with hat on top minus 11 straight k with hat on top over denominator square root of open parentheses negative 1 close parentheses squared plus open parentheses 6 close parentheses squared plus open parentheses negative 11 close parentheses squared end root end fraction equals fraction numerator negative straight i with hat on top plus 6 straight j with hat on top minus 11 straight k with hat on top over denominator square root of 158 end fraction end style

Q18. The vectors

are the diagonal of a parallelogram. What is the area of the parallelogram?

Solution

If are the diagonal of a parallelogram, then the area of the parallelogram is

vertical line a with rightwards arrow on top x b with rightwards arrow on top vertical line

Q19. The position vectors of A, B and C are

and

. Prove that A, B and C are collinear.

Solution

Here, we have Since,, therefore are parallel, but B is the common point in both the vectors. Thus, A, B and C are collinear.

Q20. If

, then find the value of

Solution

Here, we have

Q21. Find unit vector in the direction of vector

Solution

Q22.

Solution

Q23. If

, then find the vector

, such that

Solution

Here, we have Let On solving the equations, we get

12 NCERT MATH MOST IMPORTANT QUESTIONS

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