MediumMathematicsHigh School (11–12)Vectors and Three-Dimensional GeometryNumerical

Two struts of a frame point along the vectors a = i + 2 j + 2 k and b = 2 i - 2 j + k…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Two struts of a frame point along the vectors $\vec a = \hat i + 2\hat j + 2\hat k$ and $\vec b = 2\hat i - 2\hat j + \hat k$ . A new support must stand at right angles to both struts. The designer needs the size of $\vec a\times\vec b$ , and then a support of length $6$ metres along $\vec a\times\vec b$ . Give the $x$ -component and the $z$ -component of this support vector.

Need a reminder? The key ideaShow

The cross product is the determinant

\displaystyle \vec a\times\vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}.

It is at right angles to both $\vec a$ and $\vec b$ , and $\vec b\times\vec a = -\vec a\times\vec b$ . For a unit vector at right angles to both, divide it by its own size.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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