Q1.6 Unit vectors and dot products (S)
The figure shows a cube of side a with edges lying along rectangular axes x, y, z. |
- Write down expressions for the vectors
and
in terms of unit vectors î,
and
.
- Using these expressions, and the properties of unit vectors,
deduce the value of
.
- Appealing to the definition of the dot-product, deduce the value of
the angle between the body diagonal (
) and the edge (
) of the cube.
Hint
Reveal
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Hint
You need to equate the two expressions for the dot product ie the explicit result you get by using the components, and the one that involves the angle you are looking for.Solution
Reveal
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Solution
- We can write the two vectors as follows
- The scalar product of the two vectors can thus be written as
- Since
we can also writewhere θ is the angle between the two vectors (and thus, quite generally, the angle between a body-diagonal and an edge). Equating the two expressions for the dot product we have