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Find the angle between two distinct diagonals of a cube.

Short Answer

Expert verified

The angle between two distinct diagonals of a cube is \(\frac{\pi }{6}\).

Step by step solution

01

Step 1. Given Information

An angle can be defined as a space formedwhen two straight lines or rays meet at a common endpoint.

The diagonal of a cube can be defined as the line segment that joins any two non-adjacent vertices in it. In Cube, we have 12 face diagonals 2 on each face, and 4 space diagonals.

02

Step 2. Find the position vector of the diagonals

To find the angle between the two distinct diagonals of a cube. Let the coordinates of ABCDEFGH be \((0, 0, 0\)), \((1, 0, 0\)), \((1, 1, 0\)), \((0, 1, 0\)), \((0, 1, 1), (0, 0, 1\)), \((1, 0, 1\)), \((1, 1, 1\)).

Now, let's consider the two diagonals, AG and DF.

Let's find the position vector of these diagonals so,

\(\underset{AG}{\rightarrow}=\left<1-0, 0-0, 1-0 \right>\)

\(\underset{AG}{\rightarrow}=\left<1,0,1 \right>\)

And

\(\underset{DF}{\rightarrow}=\left<0-0, 0-1, 1-0 \right>\)

\(\underset{DF}{\rightarrow}=\left<0,-1,1 \right>\)

Now, the magnitude of the vectors are

\(\left\|\underset{AG}{\rightarrow} \right\|=\sqrt{1^{2}+0^{2}+1^{2}}\)

\(\left\|\underset{AG}{\rightarrow} \right\|=\sqrt{2}\)

And

\(\left\|\underset{DF}{\rightarrow} \right\|=\sqrt{0^{2}+(-1)^{2}+1^{2}}\)

\(\left\|\underset{DF}{\rightarrow} \right\|=\sqrt{2}\)

03

Step 3. Find the angle

Now, the angle between two vectors \(\underset{a}{\rightarrow}\) and \(\underset{b}{\rightarrow}\) is \(\theta=cos^{-1} \frac{\underset{a}{\rightarrow}\cdot \underset{b}{\rightarrow}}{\left| \underset{a}{\rightarrow}\right|\left|\underset{b}{\rightarrow} \right|}\) .

So,

\(\underset{AG}{\rightarrow}\cdot \underset{GF}{\rightarrow}=\left< 1,0,1\right>\cdot \left<0,-1,1 \right>\)

\(\underset{AG}{\rightarrow}\cdot \underset{GF}{\rightarrow}=1\)

Let's put all the values in the formula of the angle between two vectors

\(\theta=cos^{-1} \frac{1}{\sqrt{2}\sqrt{2}}\)

\(\theta =cos^{-1}\left ( \frac{1}{2} \right )\)

\(\theta =\frac{\pi }{6}\)

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Most popular questions from this chapter

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is k=1.

(b) True or False: โˆ‘k=0nโ€Š1k+1+โˆ‘k=1nโ€Šk2is equal to โˆ‘k=0nโ€Šk3+k2+1k+1.

(c) True or False: โˆ‘k=1nโ€Š1k+1+โˆ‘k=0nโ€Šk2is equal to โˆ‘k=1nโ€Šk3+k2+1k+1 .

(d) True or False: โˆ‘k=1nโ€Š1k+1โˆ‘k=1nโ€Šk2 is equal to โˆ‘k=1nโ€Šk2k+1.

(e) True or False: โˆ‘k=0mโ€Šk+โˆ‘k=mnโ€Škis equal toโˆ‘k=0nโ€Šk.

(f) True or False: โˆ‘k=0nโ€Šak=โˆ’a0โˆ’an+โˆ‘k=1nโˆ’1โ€Šak.

(g) True or False: โˆ‘k=110โ€Šak2=โˆ‘k=110โ€Šak2.

(h) True or False: โˆ‘k=1nโ€Šex2=exex+12ex+16.

Find u+vand u-v. Also sketchu,v,u+vand u-v.

role="math" localid="1649578020551" u=3,-4andv=-1,5

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.

u=3,-1,2,v=-4,-6,3

In Exercises 36โ€“41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(3,1,8),Q(0,6,โˆ’1),R(โˆ’3,5,โˆ’3)

Find u+vand u-v. Also, sketch u,v,u+vand u-v.

role="math" localid="1649572653771" u=2,-6,v=6,2,

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