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(ii) Write down two facts which your answers to (

(ii) Write down two facts which your answers to (

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Question

(ii) Write down two facts which your answers to (c) and (d) tell you about A,BA,B and CC.
[1]
June 2000/m/11
[1] 35p=(14),q=(34)35 p=([1],[],[-4]),q=([-3],[],[4]) and r=(m2)r=([m],[],[2]).
(a) Find |q||q|.
(b) Express 2pq2p-q as a column vector.
[1]
(c) Given that pp is parallel to rr, find mm.
[1]
[1]
[1]
Nov. 2000/1/10

Best Answer

(a) |q||q|
Answer: |q|=(3)2+42=9+16=25=5|q|=sqrt((-3)^(2)+4^(2))=sqrt(9+16)=sqrt25=5.
(b) 2pq2p-q as a column vector
Answer: 2pq=2(35140)(34)=(70280)(34)=(73284)2p-q=2([35],[],[-140])-([-3],[],[4])=([70],[],[-280])-([-3],[],[4])=([73],[],[-284]). Therefore, 2pq=(73284)2p-q=([73],[],[-284]).
© mm if pp is parallel to rr
Answer: Since pp is parallel to rr, we can write p=krvec(p)=k vec(r) for some scalar kk. Thus, (35140)=k(m2)([35],[],[-140])=k([m],[],[2]). Equating the components gives 35=km35=km and 140=2k-140=2k, which gives k=70k=-70. Substituting this into the first equation gives 35=70m35=-70 m, so m=12m=-(1)/(2). Therefore, m=12m=-(1)/(2).
2 facts which the answers to © and (d) tell us about AA, BB, and CC:
  • The vector XDvec(XD) points in the opposite direction as vector ABvec(AB). Thus, XDXD is parallel to ABAB and the opposite sides in the parallelogram ABCDABCD are parallel.
  • The vector AXvec(AX) is not parallel to vector ABvec(AB), so AXAX is not a diagonal of parallelogram ABCDABCD.