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The picture shows a model of a water molecule - Scottish Highers Maths - Question 5 - 2016
Question 5
The picture shows a model of a water molecule. Relative to suitable coordinate axes, the oxygen atom is positioned at point A(-2, 2, 5). The two hydrogen atoms are... show full transcript
Worked Solution & Example Answer:The picture shows a model of a water molecule - Scottish Highers Maths - Question 5 - 2016
Step 1
Express \( \vec{AB} \) and \( \vec{AC} \) in component form.
Answer
To express the vectors ( \vec{AB} ) and ( \vec{AC} ) in component form, we can subtract the coordinates of point A from those of points B and C.
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Calculate ( \vec{AB} ):
[ \vec{AB} = B - A = (-10, 18, 7) - (-2, 2, 5) ]
[ = (-10 + 2, 18 - 2, 7 - 5) ]
[ = (-8, 16, 2) ] -
Calculate ( \vec{AC} ):
[ \vec{AC} = C - A = (-4, -6, 21) - (-2, 2, 5) ]
[ = (-4 + 2, -6 - 2, 21 - 5) ]
[ = (-2, -8, 16) ]
Step 2
Hence, or otherwise, find the size of angle BAC.
Answer
To find the angle BAC, we first calculate the lengths of the vectors ( \vec{AB} ) and ( \vec{AC} ).
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Calculate lengths:
[ |\vec{AB}| = \sqrt{(-8)^2 + (16)^2 + (2)^2} = \sqrt{64 + 256 + 4} = \sqrt{324} = 18 ]
[ |\vec{AC}| = \sqrt{(-2)^2 + (-8)^2 + (16)^2} = \sqrt{4 + 64 + 256} = \sqrt{324} = 18 ] -
Scalar Product:
[ \vec{AB} \cdot \vec{AC} = (-8)(-2) + (16)(-8) + (2)(16) ]
[ = 16 - 128 + 32 = -80 ] -
Using cos formula:
[ \cos{BAC} = \frac{\vec{AB} \cdot \vec{AC}}{|\vec{AB}| |\vec{AC}|} = \frac{-80}{18 \times 18} = \frac{-80}{324} ] -
Angle BAC:
[ BAC = \cos^{-1}\left( \frac{-80}{324} \right) \approx 104.3^{\circ} \text{ or } 1.82 \text{ radians} ]