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

Two vectors a and b have the sizes 3 and 5, and their dot product is 9…

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

Two vectors $\vec a$ and $\vec b$ have the sizes $3$ and $5$ , and their dot product is $9$ . A designer wants the size of $\vec a\times\vec b$ without finding any components. She then uses it for a parallelogram frame made on the two vectors, and for a triangular sail made on the same vectors. Find the area of the parallelogram, the area of the triangle and $\sin\theta$ , where $\theta$ is the angle between the vectors.

Need a reminder? The key ideaShow

The area of the parallelogram on $\vec a$ and $\vec b$ is $\lvert\vec a\times\vec b\rvert$ , and the triangle has half of that. Also $\lvert\vec a\times\vec b\rvert^2 = \lvert\vec a\rvert^2\lvert\vec b\rvert^2 - (\vec a\cdot\vec b)^2$ .

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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