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Pre-Calc Homework Solutions 408

# Pre-Calc Homework Solutions 408 - 408 Section 10.2 40 The...

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35. continued (d) M is a fraction of the way from P to Q . Let d be this fraction. Then E OM 5 d E OQ 1 (1 2 d ) E OP . Proof: E PM 5 d E PQ and E MQ 5 (1 2 d ) E PQ , so E PQ 5 } 1 d } E PM and E PQ 5 } 1 2 1 d } E MQ . Therefore, } 1 d } E PM 5 } 1 2 1 d } E MQ . But E PM 5 E OM 2 E OP and E MQ 5 E OQ 2 E OM , so } 1 d } E OM 2 } 1 d } E OP 5 } 1 2 1 d } E OQ 2 } 1 2 1 d } E OM . Therefore, } 1 d } E OM 1 } 1 2 1 d } E OM 5 } 1 d } E OP 1 } 1 2 1 d } E OQ . E OM 1 } d (1 1 2 d ) } 2 5 } 1 d } E OP 1 } 1 2 1 d } E OQ E OM 5 (1 2 d ) E OP 1 d E OQ . 36. E CA 5 2 u 2 v and E CB 5 u 2 v . Since ) v ) 5 ) u ) , these vectors are orthogonal, as ( 2 u 2 v ) ? ( u 2 v ) 5 ) v ) 2 2 ) u ) 2 5 0. 37. Two adjacent sides of the rhombus can be given by two vectors of the same length, u and v . Then the diagonals of the rhombus are ( u 1 v ) and ( u 2 v ). These two vectors are orthogonal since ) u ) 5 ) v ) so ( u 1 v ) ? ( u 2 v ) 5 ) u ) 2 2 ) v ) 2 5 0. 38. Two adjacent sides of a rectangle can be given by two vec- tors u and v . The diagonals are then ( u 1 v ) and ( u 2 v ). These two vectors will be orthogonal if and only if u and v are the same length, since ( u 1 v ) ?
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