Matematikkens mysterier 4. Rumgeometri - KennethHansen.net
Matematikkens mysterier 4. Rumgeometri - KennethHansen.net
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⎠