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Let A B C D be a p[arallelogram whose di...

Let `A B C D` be a p[arallelogram whose diagonals intersect at `P` and let `O` be the origin. Then prove that ` vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot`

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Since the diagonals of a parallelogram bisect each other, P is the middle point of AC and BD both. Therefore,
`vec(OA) + vec(OC) = 2 vec(OP) and vec(OB) + vec(OD) = 2 vec(OP)`
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CENGAGE PUBLICATION-INTRODUCTION TO VECTORS -CONCEPT APPLICATION EXERCISE 1.1
  1. Find the unit vector in the direction of the vector veca=hati+hatj+2ha...

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  2. Find the direction cosines of the vector hati+2hatj+3hatk.

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  3. Find the direction cosines of the vector joining the points A(1, 2, -3...

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  4. The position vectors of P and Q are 5hati+4hatj+ahatk and -hati+2hatj-...

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  5. Given three points are A(-3,-2,0),B(3,-3,1)a n dC(5,0,2)dot Then find ...

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  6. Find a vector of magnitude 5 units, and parallel to the resultant of t...

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  7. Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are col...

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  8. If A B C D is a rhombus whose diagonals cut at the origin O , then ...

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  9. Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , re...

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  10. Let A B C D be a p[arallelogram whose diagonals intersect at P and ...

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  11. If A B C D is quadrilateral and Ea n dF are the mid-points of A Ca n d...

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  12. If vec A O+ vec O B= vec B O+ vec O C , then A ,Bn a dC are (where O ...

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  13. If the sides of an angle are given by vectors veca=hati-2hatj+2hatk an...

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  14. A B C D is a parallelogram. If La n dM are the mid-points of B Ca n dD...

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  15. A B C D is a quadrilateral and E and the point intersection of the ...

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  16. What is the unit vector parallel to vec a=3 hat i+4 hat j-2 hat k ...

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  17. The position vectors of points A and B w.r.t. the origin are veca=hati...

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  18. If vec r1, vec r2, vec r3 are the position vectors off thee collinear...

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  19. If veca and vecb are two vectors of magnitude 1 inclined at 120^(@), t...

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  20. Find the vector of magnitude 3, bisecting the angle between the vector...

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