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Prove that vectors vec u=(a l+a1l1) hat...

Prove that vectors ` vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k` ` vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k` ` vec w=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k`

A

form an equilateral triangle

B

are coplanar

C

are collinear

D

are mutually perpendicular

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The correct Answer is:
B
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vec a = a_1 hat i a_2 hat j a_3 hat k |a| = 1 , &, vec a. vec b = 2 vec b = b_1 hat i b_2 hat j b_3 hat k |b| = 4 vec c = 2 (vec a times vec b) - 3 vec b then angel between vec c and vec b

The equation of the line passing through the points a_1 hat i+a_2 hat j+a_3 hat k and b_1 hat i+b_2 hat j+b_3 hat k is -> r=(a_1 hat i+a_2 hat j+a_3 hat k)+lambda(b_1 hat i+b_2 hat j+b_3 hat k) -> r=(a_1 hat i+a_2 hat j+a_3 hat k)-t(b_1 hat i+b_2 hat j+b_3 hat k) -> r=a_1(1-t) hat i+_2(1-t) hat j+a_3(1-t) hat k+t(b_1 hat i+b_2 hat j+b_3 hat k)dot None of these

If vec a and vec b are non-zero and non-collinear vectors, then vec ax vec b=[ vec a vec b hat i] hat i+[ vec a vec b hat j] hat j+[ vec a vec b hat k] hat k vec adot vec b=( vec adot vec i)( vec adot hat i)( vec bdot hat j)+( vec adot hat j) ( vec bdot hat j) + ( vec adot hat k)( vec bdot hat k) If vec u= hat a-( hat adot hat b) hat b and hat v= hat ax hat b , then | vec v|=| vec u| If vec c= vec ax( vec ax vec b) , then vec cdot vec a=0

Let V be the volume of the parallelepiped formed by the vectors vec a=a_(i)hat i+a_(2)hat j+a_(3)hat k and vec b=b_(1)hat i+bhat j+b_(3)hat k and vec c=c_(1)hat i+c_(2)hat j+c_(3)hat k. If a_(r),b_(r) and cr, where r=1,2,3, are non-negative real numbers and sum_(r=1)^(3)(a_(r)+b_(r)+c_(r))=3L show that V<=L^(3)

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Let vec a = a_1 hat i + a_2 hat j+ a_3 hat k;vec b = b_1 hat i+ b_2 hat j+ b_3 hat k ; vec c= c_1hat i + c_2 hat j+ c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a & vec b. If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(a) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(a) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(c ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(n) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}| is equal to

ARIHANT MATHS-PRODUCT OF VECTORS-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that vectors vec u=(a l+a1l1) hat i+(a m+a1m1) hat j+(a n+a1n1)...

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  2. Let O be the origin and let PQR be an arbitrary triangle. The point S ...

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  3. Let O be the origin and OX, OY, OZ be three unit vectors in the direct...

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  4. Let O be the origin, and O X , O Y , O Z be three unit vectors ...

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  5. Let a, b and c be three unit vectors such that atimes(btimesc)=(sqrt(3...

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  6. Let vec(a) , vec(b) and vec(c) be three non-zero vectors such that no ...

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  7. If a, b and c are unit vectors satisfying |a-b|^(2)+|b-c|^(2)+|c-a|^(2...

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  8. The vector(s) which is/are coplanar with vectors hat(i)+hat(j)+2hat(k)...

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  9. Let vec a =hat i+hat j+hat k,vec b=hat i -hat j+hat k and vec c =hat ...

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  10. Two adjacent sides of a parallelogram ABCD are given by vec(AB)=2hati+...

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  11. Let P, Q, R and S be the points on the plane with position vectors -2h...

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  12. If a and b are vectors in space given by a=(hat(i)-2hat(j))/(sqrt(5)) ...

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  13. If veca,vecb,vecc and vecd are unit vectors such that (vecaxxvecb)*(...

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  14. The edges of a parallelopiped are of unit length and are parallel to ...

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  15. Lelt two non collinear unit vectors hata and hatb form and acute angle...

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  16. Let the vectors PQ, QR, RS, ST, TU and UP represent the sides of a reg...

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  17. The number of distinct real values of lambda, for which the vectors -l...

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  18. Let veca,vecb,vecc be unit vectors such that veca+vecb+vecc=vec0. Whic...

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  19. Let A be vector parallel to line of intersection of planes P1 and P2. ...

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  20. Let a=hat(i)+2hat(j)+hat(k), b=hat(i)-hat(j)+hat(k), c=hat(i)+hat(j)-h...

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  21. If vec a , vec b and vec c are three non-zero, non coplanar vecto...

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