Vectors: unknown components

**veralix** · February 12th 2009, 05:32 AM

I have this question: "If vector c forms an obtuse angle with the OY-axis and is orthogonal to both vectors a = 3i + 2j + 2k and b = 18i - 22j - 5k. Find the component of c if it's norm is 14"

This I can gather:

a dot c = 0 (because they are orthogonal)
b dot c = 0 (because they are orthogonal)
a x b = 0 (because both a and b are orthogonal to c, a and b must be paralell)
|a| = sqrt(17)
|b| = 7sqrt(17)
|a x c| = 119
|a x b| 98sqrt(17)

I know from the equation that if the angle is obtuse with the 0Y axis, the Y component must be negative, so I surmise that c = c1 *i - c2 *j + c3*k

Other than that I'm completely lost, I can't figure out how to find the components of vector c

**Plato** · February 12th 2009, 07:23 AM

Originally Posted by veralix

I have this question: "If vector c forms an obtuse angle with the OY-axis and is orthogonal to both vectors a = 3i + 2j + 2k and b = 18i - 22j - 5k. Find the component of c if it's norm is 14"

First $\;b \times a = \left\langle { - 34, - 51, + 102} \right\rangle \;\& \;\left\| {b \times a} \right\| = 119$ perpendicular to both making obtuse angle with j.
Make a unit vector and multiply by 14: $\left\langle {\frac{{ - 476}}<br /> {{119}},\frac{{ - 714}}<br /> {{119}},\frac{{1428}}<br /> {{119}}} \right\rangle \;$

**HallsofIvy** · February 12th 2009, 08:40 AM

Or:
Writing c as $c_1\vec{i}+ c_2\vec{j}+ c_3\vec{k}$ , knowing that $\vec{a}$ and $\vec{b}$ are orthogonal, you know that $\vec{a}\cdot\vec{c}= 3c_1+ 2c_2+ 2c_2= 0$ and knowing that $\vec{b}$ and $\vec{c}$ are othogonal, you know that $18c_1- 22c_2- 5c_3= 0$ . Knowing that $|\vec{c}|= 14$ you know that $c_1^2+ c_2^2+ c_3^2= 196$ . You can solve the first two, linear, equations for $c_1$ and $c_2$ as functions of $c_3$ and put those into the last equation to get a quadratic equation for $c_3$ . It's the fact that that quadratic equation has two solutions that requires that additiona information about the angle.

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