Can u pls..solve questions from 46 - 8

Can u pls..solve questions from 46 - 8 46. The sum and difference of two vectors are perpendicular to other. Prove that the vectors are equal in magnitude. 47.Find the conditions for two vectors to be (i) parallel and (iil perpendicular to each other. 48. The resultant of two vectors is perpendicular to mind i magnitude is half that of Q. What is the angle between i'&Ä? (PROBLEMS INVOLVING UNIT VECTORS 7,} & 1. Find a unit vector parallel to the resultant of the vectors 4j 2k and B = 3i-5j + k. 2. Show that vectors = 2i — 3j —k and = -6i + 9j +3k are parallel each other. 3. Find the magnitude and direction Of (i) i + j and (ii) i —j. . A vector OP has its initial point O at (2, 2,3) and terminal poi 4 5. (-3, 1, -2). Find the equation of the vector. In a Cartesian co-ordinate system, the co-ordinate points•aie 3, -6). If a point is given a displacement -4i — 8j + 13k, wh@tyil the co-ordinates of the new position? 6. Three forces = 2i + j — k, = 2i + 3j -3k and = a (i Newton, act simultaneously on a particle and ke•eø: i equilibrium. Find the value of 'a' 7. Find the vector parallel to the resultant of vector = 2i + 4j -4.6k 8. What is the nature of the triangle formed by the vectors 2i*-3j 3j+j-2kand 5i 3k?

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L e t t h e t w o v e c t o r s b e \vec{a} a n d \vec{b} . G i v e n t h e s u m a n d d i f f e r e n c e o f t h e t w o v e c t o r s a r e p e r p e n d i c u l a r t o e a c h o t h e r . \Rightarrow (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 0 \Rightarrow \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} = 0 \Rightarrow \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} = 0 (∵ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}) \Rightarrow {|\vec{a}|}^{2} - {|\vec{b}|}^{2} = 0 \Rightarrow {|\vec{a}|}^{2} = {|\vec{b}|}^{2} \Rightarrow |\vec{a}| = |\vec{b}| H e n c e p r o v e d

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