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Matematikkens mysterier 4. Rumgeometri - KennethHansen.net

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1.1 a)<br />

⎛ 5 ⎞<br />

⎜ ⎟<br />

⎜ 4 ⎟<br />

⎜ ⎟<br />

⎝14⎠<br />

b)<br />

⎛ 5 ⎞<br />

⎜ ⎟<br />

⎜−10⎟<br />

⎜ ⎟<br />

⎝ −4<br />

⎠<br />

Facitter<br />

c)<br />

g) 53 h) 58, 34°<br />

i)<br />

⎛−10⎞<br />

⎜ ⎟<br />

⎜ −7<br />

⎟<br />

⎜ ⎟<br />

⎝−22⎠<br />

⎛−6<br />

/ 7⎞<br />

⎜ ⎟<br />

⎜ 3/ 7 ⎟<br />

⎜ ⎟<br />

⎝12<br />

/ 7⎠<br />

1.2 a) 6 b) 0° c) 180°<br />

1.3 a) 1 ( 1± 5)<br />

b) ± 11<br />

2<br />

52<br />

d) 14 e) 21 f) 9<br />

j) 124, 57°<br />

k)<br />

⎛−27⎞<br />

⎜ ⎟<br />

⎜−81⎟<br />

⎜ ⎟<br />

⎝−54⎠<br />

1.4 b) AB = 20 AC = 22 BC = 14<br />

∠ A = 48, 13°<br />

∠ B = 68, 99°<br />

∠ C = 62, 88°<br />

c) ( 4 , 2 , − 2)<br />

d) ( 5 , 3<br />

2 2 , − 1)<br />

og ( 2 , 4<br />

3 , 0)<br />

1.5 a) B = ( 6 , 1, 3 ) C = ( 3 , 3 , 3 )<br />

2.1 a)<br />

3.1 a)<br />

3.2 a)<br />

b) AB = 41 AC = 18 BC = 13<br />

∠ A = 32, 15°<br />

∠ B = 38, 77°<br />

∠ C = 109, 08°<br />

c) ( −1, 6 , −1<br />

)<br />

⎛ 4 ⎞<br />

⎜ ⎟<br />

⎜ 1 ⎟<br />

⎜ ⎟<br />

⎝−6⎠<br />

b)<br />

⎛0⎞<br />

⎜ ⎟<br />

⎜0⎟<br />

⎜ ⎟<br />

⎝0⎠<br />

g) 289 h) 1<br />

2<br />

c)<br />

⎛−4⎞<br />

⎜ ⎟<br />

⎜−1⎟<br />

⎜ ⎟<br />

⎝ 6 ⎠<br />

d)<br />

⎛11⎞<br />

⎜ ⎟<br />

⎜16⎟<br />

⎜ ⎟<br />

⎝10⎠<br />

441 i) 137, 70°<br />

j) nej<br />

e)<br />

⎛11⎞<br />

⎜ ⎟<br />

⎜16⎟<br />

⎜ ⎟<br />

⎝10⎠<br />

⎛ x⎞<br />

⎜ ⎟<br />

⎜ y⎟<br />

s t<br />

⎜ ⎟<br />

⎝ z⎠<br />

=<br />

⎛ ⎞<br />

⎜ ⎟<br />

⎜ ⎟<br />

⎜ ⎟<br />

⎝ ⎠<br />

+<br />

⎛−<br />

⎞<br />

⎜ ⎟<br />

⎜ ⎟<br />

⎜ ⎟<br />

⎝ − ⎠<br />

+<br />

0 9 / 2 ⎛ 0 ⎞<br />

⎜ ⎟<br />

0 0 ⎜ 3 ⎟<br />

⎜ ⎟<br />

9 9 ⎝−9⎠<br />

c) 16x − 18y − 3z + 50 = 0<br />

d) 102, 08°<br />

e) nej<br />

3.3 a) linien<br />

⎛ x⎞<br />

⎜ ⎟<br />

⎜ y⎟<br />

t<br />

⎜ ⎟<br />

⎝ z⎠<br />

=<br />

⎛8⎞<br />

⎛15⎞<br />

⎜ ⎟ ⎜ ⎟<br />

⎜3⎟<br />

+ ⎜−1⎟<br />

⎜ ⎟ ⎜ ⎟<br />

⎝4⎠<br />

⎝−8⎠<br />

c) linien<br />

⎛ x⎞<br />

⎜ ⎟<br />

⎜ y⎟<br />

t<br />

⎜ ⎟<br />

⎝ z⎠<br />

=<br />

⎛ 0 ⎞ ⎛ 0 ⎞<br />

⎜ ⎟ ⎜ ⎟<br />

⎜ 3/ 7 ⎟ + ⎜−1/<br />

7⎟<br />

⎜ ⎟ ⎜ ⎟<br />

⎝−3<br />

/ 2⎠<br />

⎝−1/<br />

2⎠<br />

⎛ x⎞<br />

⎜ ⎟<br />

⎜ y⎟<br />

t<br />

⎜ ⎟<br />

⎝ z⎠<br />

=<br />

⎛ − ⎞<br />

⎜ ⎟<br />

⎜ − ⎟<br />

⎜ ⎟<br />

⎝−<br />

⎠<br />

+<br />

12 ⎛120⎞<br />

⎜ ⎟<br />

11 ⎜ 60 ⎟<br />

⎜ ⎟<br />

305 , ⎝195⎠<br />

b) 12x + 18y −11z − 107 = 0<br />

b) nej<br />

l)<br />

f)<br />

b) tom - planerne er parallelle<br />

55, 12°<br />

⎛ −2,<br />

1⎞<br />

⎜ ⎟<br />

⎜ 0, 9 ⎟<br />

⎜ ⎟<br />

⎝−2,<br />

4⎠<br />

⎛ 4 ⎞<br />

⎜ ⎟<br />

⎜ 1 ⎟<br />

⎜ ⎟<br />

⎝−6⎠

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