A unit vector perpendicular to the lines r ¯ = ( − i ¯ − 2 j ¯ − k ¯ ) + t ( 3 i ¯ + j ¯ + 2 k ¯ ) , r = ( 2 i ¯ − 2 j ¯ + 3 k ¯ ) + s ( i ¯ + 2 j ¯ + 3 k ¯ ) where s ∈ R , t ∈ R , is
Let the volume of a parallelepiped whose coterminous edges are given by u = i ^ + j ^ + λ k ^ , v = i ^ + j ^ + 3 k ^ , and w = 2 i ^ + j ^ + k ^ , be 1 cu.unit. If θ be the angle between the edges u and w , then cos θ can be:
if a and b are prependicular unit vector and vector c i s s u c h t h a t c = a + b ,then the value of a × b . b × c + b × c . c × a + c × a . a × b
If a = x i + ( x − 1 ) j + k and b = ( x + 1 ) i + j + a k always make an acute angle for all x ∈ R ,then the least integral value of a is
Let a , b and c be three unit vectors such that a + b + c = 0 . If λ = a . b + b . c + c . a and then d = a × b + b × c + c × a the ordered pair, λ , d is equal to:
If x and y are two non-collinear vectors and a , b , c represent the sides of a △ A B C satisfying ( a − b ) x + ( b − c ) y + ( c − a ) ( x × y ) = 0 , then Δ A B C is:
The volume of the tetrahedron formed by the coterminus edges a , b , c is 3 . Then the volume of the parallelepiped formed by the coterminus edges a + b , b + c , c + a is
Let a , b , c be distinct non-negative numbers. If the vectors a i ^ + a j ^ + c k ^ , i ^ + k ^ and c i ^ + c j ^ + b k ^ lie in a plane, then c is :
In the isosceles triangle ABC , | A B | = | B C | = 8 , a point E divides AB internally in the ratio 1 : 3 , then the cosine of the angle between C E and C A is : (where | C A | = 12 )
Let G 1 , G 2 , G 3 be the centroids of the triangular faces OBC , OCA , OAB of a tetrahedron OABC . If V 1 denote the volume of the tetrahedron OABC and V 2 that of the parallelopiped with O G 1 , O G 2 , O G 3 as three concurrent edges, then :
A vector of magnitude 5 5 coplanar with vectors i ^ + 2 j ^ and j ^ + 2 k ^ and the perpendicular vector 2 ^ i + j ^ + 2 k ^ is :
If a and b are two unit vectors such that a + 2 b and 5 a − 4 b are perpendicular to each other then the angle between a and b is
The vectors a = i ^ + j ^ + m k ^ , b = i ^ + j ^ + ( m + 1 ) k ^ and c = i ^ − j ^ + m k ^ are coplanar if m is =
Given a parallelogram ABCD . If | A B ∣ = a , | A D ∣= b and | A C | = c , then D B ⋅ A B has the value :
Let three vectors a , b and c be such that c is coplanar with a and b , a ⋅ c = 7 and b is perpendicular to c , where a = − i ^ + j ^ + k ^ and b = − 2 i ^ + k ^ , then the value of 2 | a + b + c | 2 is
If a 2 + b 2 + c 2 = 1 where a , b , c ∈ R , then the maximum value of ( 4 a − 3 b ) 2 + ( 5 b − 4 c ) 2 + ( 3 c − 5 a ) 2 is
If a = 5 i ^ − j ^ + k ^ , b = 2 i ^ − 3 j ^ − k ^ , c = − 3 i ^ + j ^ + k ^ and d = 2 j ^ + k ^ , then the value of d ⋅ ( a × { b × ( c × d ) } ) equals
If a , b , c are unit vectors such that a ⋅ b = 0 = a ⋅ c and the angle between b and c is π 3 . Then the value of | a × b − a × c | is
If A 1 , 2 , 3 , B 2 , 3 , 4 , C 3 , 4 , 5 , D 4 , 5 , 6 are four points, then the projection of B D ¯ on A C ¯ is
If the vertices of a triangle are 1 , 2 , 3 , 3 , 4 , − 1 , 5 , − 4 , 2 and the area of triangle is λ , then the number of positive integral divisors of λ 2 6 + λ 2 + 25 3 is
The area of the triangle formed by the points A ( 1 , 2 , 3 ) B ( – 2 , 1 , 4 ) , C ( 3 , 4 , – 2 )
A vector a = α i ^ + 2 j ^ + β k ^ , α , β ∈ R lies in the plane of the vectors b = i ^ + j ^ and c = i ^ − j ^ + 4 k ^ . If a bisects the angle between b and c , . then
The projection of the line segment joining the points 1 , − 1 , 3 and 2 , − 4 , 11 on the line joining the points − 1 , 2 , 3 and 3 , − 2 , 10 is
If the vectors, p = ( a + 1 ) i ^ + a j ^ + a k ^ , q = a i ^ + ( a + 1 ) j ^ + a k ^ and r = a i ^ + j ^ + ( a + 1 ) k ^ ( a ∈ R ) are coplanar and 3 ( p · q ) 2 – λ | r × q ¯ | 2 = 0 , then the value of λ is
The edges of a parallelopied are of unit length and are parallel to non-coplanar unit vectors a ¯ , b ¯ , c ¯ such that a ¯ b ¯ = b ¯ · c ¯ = c ¯ · a ¯ = 1 2 . Then the volume of the parallelopiped is
If a , b , and c are unit vectors such that a + 2 b + 2 c = 0 , then | a × c | is equal to
Let a = 2 i ^ – j ^ + k ^ , b = i ^ + 2 j ^ – k ^ and c = i ^ + j ^ – 2 k ^ . A vector in the plane of b and c , whose projection on a is of magnitude 2 / 3 is
Let A B C D be a tetrahedron such that the edges A B , A C and A D are mutually perpendicular. Let the area of triangles ABC , ACD and ADB be 3 , 4 and 5 sq. units respectively. Then the area of the triangle BCD is :
If the vectors a = 3 i ^ + j ^ – 2 k ^ , b = – i ^ + 3 j ^ + 4 k ^ and c = 4 i ^ – 2 j ^ – 6 k ^ constitute the sides of a ∆ A B C , then the length of the median bisecting the vector c is
Let A and B be two non-parallel unit vectors in a plane. If ( α A + B ) bisects the internal angle between A and B , then ∝ is equal to
The altitude of parallelopiped whose three coterminous edges are the vectors, A = i ^ + j ^ + k ^ ; B = 2 i ^ + 4 j ^ − k ^ and C = i ^ + j ^ + 3 k ^ with A and B as the sides of the base of the parallelopipied, is
a and b are two given vectors. On these vectors as adjacent sides, a parallelogram is constructed. The vector which is the altitude of the parallelogram and perpendicular to a is given by:
G is the centroid of the triangle ABC and if G , is the centroid of another triangle A 1 B 1 C 1 , then value of A A 1 + B B 1 + C C 1 is :
Let a , b and c be three non-zero vectors, no two of which are collinear. If the vector a + 2 b is collinear with c , and b + 3 c is collinear with , a then a + 2 b + 6 c is equal to
The values of k for which the points A ( 1 , 0 , 3 ) , B ( − 1 , 3 , 4 ) , C ( 1 , 2 , 1 ) and D ( k .2 , 5 ) are coplanar, are
A vector , c directed along the internal bi-sector of the angle between the vectors a = 7 i ^ − 4 j ^ − 4 k ^ and b = − 2 i ^ – j ^ + 2 k ^ , with | c | = 5 6 , is
Consider Δ A B C with A ≡ ( a ) ; B ≡ ( b ) and C ≡ ( c ) . If b ⋅ ( a + c ) = b ⋅ b + a ⋅ c ; | b − a | = 3 ; | c − b | = 4 then the angle between the medians A M and B D is :
u , v , w be such that | u | = 1 , | v | = 2 , | w | = 3 . If the projection of v along u is equal to that of w along and u and vectors v , w are perpendicular to each other, then | u − v + w | equals :
If a , b , c are three non-coplanar and p , q , r are reciprocal vectors to a , b and c respectively, then l a + m b + n c ⋅ l p + m q + n r is equal to : (where 1 , m , n are scalars)
Let a ¯ , b ¯ and c ¯ be the unit vectors such that a ¯ and b ¯ are mutually perpendicular and c ¯ is equally inclined to a ¯ and b ¯ at angle θ . If c ¯ = x a ¯ + y b ¯ + z ( a ¯ × b ¯ ) , then
If the angle between the vectors λ i ^ − 3 j ^ + 5 k ^ and 2 λ i ^ − λ j ^ − k ^ is an obtuse angle and λ is an integer, then the number of value(s) of λ is/are
Let u ⇀ b e a v e c t o r o n r e c t a n g u l a r c o o r d i n a t e s y s t e m w i t h s l o p i n g a n g l e 60 0 , s u p p o s e t h a t u – i ^ i s g e o m e t r i c m e a n o f u a n d u – 2 i ^ w h e r e i ^ i s t h e u n i t v e c t o r a l o n g x – a x i s t h e n t h e v a l u e o f ( 2 + 1 ) u i s
The altitude of a parallelepiped whose three coterminous edges are vectors a ¯ = i ¯ + j ¯ + k ¯ , b ¯ = 2 i ¯ + 4 j ¯ − k ¯ and c ¯ = i ¯ + j ¯ + 3 k ¯ with a ¯ and b ¯ as sides of the base of the parallelopiped is
Let A ¯ = i ¯ − 3 j + 4 k ¯ , B ¯ = 6 i ¯ + 4 j − 8 k , C ¯ = 5 i ¯ + 2 j ¯ + 5 k ¯ and a vector R ¯ satisfies R ¯ × B ¯ = C ¯ × B ¯ , R ¯ ⋅ A ¯ = 0 then value of | B ¯ | | R ¯ − C ¯ | is
A parallelogram is constructed on 5 a + 2 b and a − 3 b as adjacent sides, where | a | = 2 2 and | b | = 3 & angle between a & b is π 4 . If the magnitude of longer diagonal is α β γ where α β γ is three digit number then α + β − γ is equal to
Let a ¯ , b ¯ , c ¯ be three vectors such that | a | = 1 , | b ¯ | = 1 and | c ¯ | = 2 and if a ¯ × ( a ¯ × c ¯ ) + b ¯ = 0 ¯ , then angle between a ¯ and c ¯ can be
Let a ¯ , b ¯ , c ¯ be three unit vectors such that a ¯ + 5 b ¯ + 3 c ¯ = 0 ¯ then a ¯ ⋅ ( b ¯ × c ¯ ) is equal to
If a and b are two vectors such that | a | = 1 , | b | = 4 , a · b = 2 and c = ( 2 a × b ) − 3 b then the angle between b and c is
Line r = a + λ b will not meet the plane r ⋅ n = q , if
If a , b are vectors perpendicular to each other and | a | = 2 , | b | = 3 , c × a = b , then the least value of | c − a | is
