If u is a unit vector, find u · v and u · w. (assume v and w are also unit vectors.) Equilateral triangle.

Solution:

Given, u is a unit vector.

v and w are also unit vectors.

u, v and w form an equilateral triangle.

We have to find u.v and u.w

By the definition of dot product,

The scalar product \(\overrightarrow{u}.\overrightarrow{v}=\left \| \overrightarrow{u} \right \|.\left \| \overrightarrow{v} \right \|.cos\alpha\) --- (1)

\overrightarrow{A}

\(\overrightarrow{u}.\overrightarrow{w}=\left \| \overrightarrow{u} \right \|.\left \| \overrightarrow{w} \right \|.cos(360^{\circ}-\alpha)\) --- (2)

Where, u, v, w are unit vectors.

α is the internal angle of the equilateral triangle.

Equilateral triangles are triangles whose sides have the same length and all internal angles have the same measure.

We know that the sum of all the internal angles of a triangle is always equal to 180 degrees.

The measure of each internal angle in an equilateral triangle equals 60 degrees.

We know,

\(\left \| \vec{u} \right \|=\left \| \vec{v} \right \|=\left \| \vec{w} \right \|=1\)

α = 60°

So, \(\overrightarrow{u}.\overrightarrow{v}=(1).(1).cos60^{\circ}\)

\(\overrightarrow{u}.\overrightarrow{v}=(1).(1).(\frac{1}{2})\)

\(\overrightarrow{u}.\overrightarrow{v}=\frac{1}{2}\)

\(\overrightarrow{u}.\overrightarrow{w}=(1).(1).cos(360^{\circ}-60^{\circ})\)

\(\overrightarrow{u}.\overrightarrow{w}=cos300^{\circ}\)

\(\overrightarrow{u}.\overrightarrow{w}=(\frac{1}{2})\)

Therefore, u.v = 1/2 and u.w = 1/2.

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If u is a unit vector, find u · v and u · w. (assume v and w are also unit vectors.) Equilateral triangle.

Summary:

If u is a unit vector, then u · v is 1/2 and u · w. = 1/2 (assume v and w are also unit vectors.) Equilateral triangle.