DOT PRODUCT PROBLEMS AND SOLUTIONS

(1)  Find a vector . b vector when

(i) a vector = i vector−2 vector+k vector and

b = 3i vector  − 4j vector−2k vector

(ii) a = 2i vector + 2j vector − k vector and

b = 6i vector − 3j vector + 2k vector.       Solution

(2)  Find the value λ are perpendicular, where

(i) a = 2i vector + λj vector + k vector and

b = i vector − 2j vector  + 3k vector

(ii) a = 2i vector + 4j vector − k vector and

b = 3i vector − 2j vector +  λ k vector     Solution

(3)  If a vector and b vector are two vectors such that |a| = 10,|b| = 15 and a . b = 75√2, find the angle between a and b.         Solution

(4)  Find the angle between the vectors

(i) 2i vector + 3j vector − 6k vector and 6i vector − 3j vector + 2k vector

(ii) i vector − j vector and j vector − k vector.

Let a vector  =  i vector − j vector and 

b vector  =  j vector − k vector        Solution

(5)  If a vector, b vector, c vector are three vectors such that a vector + 2b vector + c vector = 0 vector and |a|= 3, |b| = 4, |c| = 7,  find the angle between a vector and b vector.         Solution

(6)  Show that the vectors a = 2i + 3j + 6k, b = 6i + 2j − 3k, and c = 3i − 6j + 2k are mutually orthogonal.    Solution

(7)  Show that the vectors −i − 2j − 6k,  2i − j + k, and − i + 3j + 5k form a right angled triangle    Solution

(8)  If | a vector| = 5, | b vector| = 6, | c vector | = 7 and a  vector + b vector + c vector  = 0, find a ⋅ b + b ⋅ c + c ⋅ a .    Solution

(9)  Show that the points (2, - 1, 3), (4, 3, 1) and (3, 1, 2) are coplanar.      Solution

(10)  If a vector, b vector are unit vectors and θ is the angle between them, show that

(i) sin  (θ/2)  =  (1/2) |a vector - b vector|

(ii) cos (θ/2)  =  (1/2) |a vector + b vector|

(ii) tan (θ/2) = |a vector-b vector| / |a vector-b vector|

  Solution

(11)  Let a vector, b vector, c vector be three vectors such that |a vector| = 3, |b vector|= 4, |c vector|= 5 and each one of them being perpendicular to the sum of the other two, find | a vector + b vector + c vector |   Solution

(12)  Find the projection of the vector i vector + 3j vector  + 7k vector on the vector 2i vector + 6j vector  + 3k vector.    Solution

(13)  Find λ, when the projection of a = λ i + j + 4k on b = 2i + 6 j + 3k is 4 units.      Solution

(14)  Three vectors a vector, b vector and c vector are such that |a vector|= 2, |b vector| = 3,|c vector| = 4 , and a vector + b vector + c vector = 0 . Find 4 a . b + 3b . c + 3c . a.    Solution

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