HardMathematicsHigh School (11–12)Vectors and Three-Dimensional Geometry › Introduction to Three-Dimensional GeometryNumerical

A steel cube frame has side 9 m. One corner is the origin O, and the edges from it run along the…

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

A steel cube frame has side $9\ \text{m}$ . One corner is the origin $O$ , and the edges from it run along the positive axes. The opposite corner is $G(9, 9, 9)$ . A cable is fixed at the corner $V(0, 9, 0)$ and it must be attached to the diagonal rod $OG$ at the point $T$ that makes the cable as short as possible. Let $T = (9t, 9t, 9t)$ . Write $VT^2$ as a quadratic in $t$ and find the value of $t$ that makes it least. Give one coordinate of $T$ . Then give the shortest length of the cable in metres to two decimal places, and the ratio $OT : TG$ as a decimal.

Need a reminder? The key ideaShow

A cuboid with edges $a$ , $b$ , $c$ has the diagonal $\sqrt{a^2 + b^2 + c^2}$ . The distance from a point to a line is found by writing the squared distance as a quadratic in $t$ . The plane $x = k$ cuts the segment $AB$ in the ratio $(k - x_1) : (x_2 - k)$ .

The worked solution

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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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