Vectors of 2 planes

Question

We have two vectors a= 3cm b= 4cm Angle between the vectors= 60 degrees. However there is another vector that is perpendicular to the plane of a and b and c= 5cm.

Now I need to find the length of a+b+c.

I already calculated it and got 7.01cm from this $$\sqrt{3^2+4^2+5^2}$$ however in the book it says 7.87cm, I'm worried that I understood the concept wrong. Did I need to use the angles somehow?

This is the question in the book:

The vectors a and b have the length 3 cm and 4 cm, respectively, and have an angle of 60 °on. The vector c stands vertically on the plane spanned by a and b and is 5 cm long. Use a construction to determine the length of a + b + c.

And the sketch in the solution:

Answer 1

Note that

$$|a+b|^2 = (a+b)\cdot (a+b ) =a^2 + b^2 +2a\cdot b=3^2+4^2+2\cdot3\cdot4\cos60=37$$

Then

$$|a+b+c|=\sqrt{c^2+|a+b|^2}= \sqrt{5^2+37} =\sqrt{62} \approx 7.874$$

Answer 2

Without loss of generality:

${\bf a} = (3,0,0)$

${\bf b} = 4 (\cos \pi/3, \sin \pi/3, 0)$

${\bf c} = (0,0,5)$

${\bf a} + {\bf b} + {\bf c} = (5, 2 \sqrt{3}, 5)$

and $|{\bf a} + {\bf b} + {\bf c}| = 7.87401.$

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Your answer is wrong because your formula assumes the vectors are mutually perpendicular, which is explicitly not the case.