Angle of the body diagonals

Using vector dot product calculate the angle of the body diagonals of the cube.

Correct result:

A =  70.5288 °

Solution:

$$\begin{gathered} D_1=(1,1,1) \\ {D}_2=(1,1,-1) \\ {d}_1=\mathrm{\mid }{D}_1\mathrm{\mid } \\ {d}_1=\sqrt{1^2+1^2+1^2}=\sqrt{3}\doteq 1.7321 \\ {d}_2=\mathrm{\mid }{D}_2\mathrm{\mid } \\ {d}_2=\sqrt{1^2+1^2+(-1{)}^2}=\sqrt{3}\doteq 1.7321 \\ {D}_1\cdot D_2=d_1{d}_2\cos A \\ c=\cos A={\displaystyle \frac{D_1\cdot D_2}{d_1{d}_2}} \\ c={\displaystyle \frac{1\cdot 1+1\cdot 1+1\cdot (-1)}{d_1\cdot d_2}}={\displaystyle \frac{1\cdot 1+1\cdot 1+1\cdot (-1)}{1.7321\cdot 1.7321}}={\displaystyle \frac{1}{3}}\doteq 0.3333 \\ {A}_0=\arccos (c)=\arccos (0.3333)\doteq 1.231\text{ rad } \\ A=A_0{\to }^{\circ }=A_0\cdot {{\displaystyle \frac{180}{\pi }}}^{\circ }=1.231\cdot {{\displaystyle \frac{180}{\pi }}}^{\circ }=70.529^{\circ }=70.5288^{\circ }=70^{\circ }31^{\mathrm{\prime }}44\mathrm{"} \end{gathered}$$

Try another example

Showing 0 comments:

Next similar math problems:

• Perpendicular projection
  Determine the distance of a point B[1, -3] from the perpendicular projection of a point A[3, -2] on a straight line 2 x + y + 1 = 0.
• Vector v4
  Find the vector v4 perpendicular to vectors v1 = (1, 1, 1, -1), v2 = (1, 1, -1, 1) and v3 = (0, 0, 1, 1)
• Decide 2
  Decide whether points A[-2, -5], B[4, 3] and C[16, -1] lie on the same line
• Vector perpendicular
  Find the vector a = (2, y, z) so that a⊥ b and a ⊥ c where b = (-1, 4, 2) and c = (3, -3, -1)
• Vector equation
  Let’s v = (1, 2, 1), u = (0, -1, 3) and w = (1, 0, 7) . Solve the vector equation c1 v + c2 u + c3 w = 0 for variables c1 c2, c3 and decide weather v, u and w are linear dependent or independent
• Parametric form
  Calculate the distance of point A [2,1] from the line p: X = -1 + 3 t Y = 5-4 t Line p has a parametric form of the line equation. ..
• Find the 10
  Find the value of t if 2tx+5y-6=0 and 5x-4y+8=0 are perpendicular, parallel, what angle does each of the lines make with the x-axis, find the angle between the lines?
• Scalar product
  Calculate the scalar product of two vectors: (2.5) (-1, -4)
• Find the 5
  Find the equation of the circle with center at (1,20), which touches the line 8x+5y-19=0
• Three points 2
  The three points A(3, 8), B(6, 2) and C(10, 2). The point D is such that the line DA is perpendicular to AB, and DC is parallel to AB. Calculate the coordinates of D.
• Angle between vectors
  Find the angle between the given vectors to the nearest tenth of a degree. u = (-22, 11) and v = (16, 20)
• Triangle
  Plane coordinates of vertices: K[11, -10] L[10, 12] M[1, 3] give Triangle KLM. Calculate its area and its interior angles.
• Coordinates of square vertices
  The ABCD square has the center S [−3, −2] and the vertex A [1, −3]. Find the coordinates of the other vertices of the square.
• Parallel and orthogonal
  I need math help in this problem: a=(-5, 5 3) b=(-2,-4,-5) (they are vectors) Decompose the vector b into b=v+w where v is parallel to a and w is orthogonal to a, find v and w
• Cuboids
  Two separate cuboids with different orientation in space. Determine the angle between them, knowing the direction cosine matrix for each separate cuboid. u1=(0.62955056, 0.094432584, 0.77119944) u2=(0.14484653, 0.9208101, 0.36211633)