PRACTICE PROBLEMS USING CROSS PRODUCT

(1)  Find the magnitude of a vector x b vector, if a vector  =  2i vector + j vector + 3k vector and b vector  =  3i vector+ 5j vector - 2k vector          Solution

(2)  Show that

Solution

(3)  Find the vectors of magnitude 103 that are  perpendicular to the plane which contains i vector + 2j vector + k vector and i vector + 3j vector + 4k vector   Solution

(4)  Find the vectors of magnitude 103 that are  perpendicular to the plane which contains i vector + 2j vector + k vector and i vector + 3j vector + 4k vector  Solution

(5)  Find the unit vectors perpendicular to each of the vectors a vector + b vector and a vector - b vector  where a vector  =  i vector + j vector + k vector and b vector =  i vector + 2j vector + 3k vector    Solution

(6)  Find the area of the parallelogram whose two adjacent sides are determined by the vectors i vector + 2j vector + 3k vector and 3i vector − 2j vector + k vector.  Solution

(7)  Find the area of the triangle whose vertices are A(3, - 1, 2), B(1, - 1, - 3) and C(4, - 3, 1).        Solution

(8)  If a vector, b vector, c vector are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is (1/2) |a × b + b × c + c × a| vector. Also deduce the condition for collinearity of the points A, B, and C.      Solution

(9)  For any vector a vector prove that

|a vector × i vector |2+|a vector × j vector|2+|a vector × k vector|2= 2 |a vector|2            Solution

(10)  Let a vector, b vector, c vector be unit vectors such that a ⋅ b = a ⋅ c = 0 and the angle between b vector and c vector is π/3. Prove that a vector  =  ± (2/3) (b × c)   Solution

(11)  Find the angle between the vectors 2i vector + j vector − k vector and i vector+ 2j vector + k vector using vector product.    Solution

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