Angle of the body diagonals

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

Result

A =  70.529 °

Solution:

D1=(1,1,1) D2=(1,1,−1)  d1=∣D1∣ d1=12+12+12=3≐1.7321  d2=∣D2∣ d2=12+12+(−1)2=3≐1.7321  D1⋅ D2=d1 d2cos⁡A  c=cos⁡A=D1⋅ D2d1d2  c=1⋅ 1+1⋅ 1+1⋅ (−1)d1⋅ d2=1⋅ 1+1⋅ 1+1⋅ (−1)1.7321⋅ 1.7321≐13≐0.3333  A0=arccos⁡(c)=arccos⁡(0.3333)≐1.231 rad  A=A0→ ∘=A0⋅ 180π  ∘=70.52878  ∘=70.529∘=70∘31′44"D_{1}=(1,1,1) \ \\ D_{2}=(1,1,-1) \ \\ \ \\ d_{1}=|D_{1}| \ \\ d_{1}=\sqrt{ 1^2+1^2+1^2 }=\sqrt{ 3 } \doteq 1.7321 \ \\ \ \\ d_{2}=|D_{2}| \ \\ 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=\dfrac{ D_{1} \cdot \ D_{2} }{ d_{1} d_{2} } \ \\ \ \\ c=\dfrac{ 1 \cdot \ 1+1 \cdot \ 1+1 \cdot \ (-1) }{ d_{1} \cdot \ d_{2} }=\dfrac{ 1 \cdot \ 1+1 \cdot \ 1+1 \cdot \ (-1) }{ 1.7321 \cdot \ 1.7321 } \doteq \dfrac{ 1 }{ 3 } \doteq 0.3333 \ \\ \ \\ A_{0}=\arccos(c)=\arccos(0.3333) \doteq 1.231 \ \text{rad} \ \\ \ \\ A=A_{0} \rightarrow \ ^\circ =A_{0} \cdot \ \dfrac{ 180 }{ \pi } \ \ ^\circ =70.52878 \ \ ^\circ =70.529 ^\circ =70^\circ 31'44"

Try another example


Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!





Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




Tips to related online calculators
For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc.
Two vectors given by its magnitudes and by included angle can be added by our vector sum calculator.
Pythagorean theorem is the base for the right triangle calculator.
See also our trigonometric triangle calculator.

Following knowledge from mathematics are needed to solve this word math problem:

Next similar math problems:

  1. Scalar dot product
    dot_product Calculate u.v if |u| = 5, |v| = 2 and when angle between the vectors u, v is: a) 60° b) 45° c) 120°
  2. Angle between vectors
    arccos Find the angle between the given vectors to the nearest tenth of a degree. u = (-22, 11) and v = (16, 20)
  3. Cuboids
    3dvectors 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)
  4. Bearing - navigation
    navigation A ship travels 84 km on a bearing of 17°, and then travels on a bearing of 107° for 135 km. Find the distance of the end of the trip from the starting point, to the nearest kilometer.
  5. Vector sum
    vectors The magnitude of the vector u is 12 and the magnitude of the vector v is 8. Angle between vectors is 61°. What is the magnitude of the vector u + v?
  6. Triangle IRT
    triangles_5 In isosceles right triangle ABC with right angle at vertex C is coordinates: A (-1, 2); C (-5, -2) Calculate the length of segment AB.
  7. Greatest angle
    triangles_4 Calculate the greatest triangle angle with sides 197, 208, 299.
  8. Vertices of RT
    RightTriangleMidpoint_3 Show that the points P1 (5,0), P2 (2,1) & P3 (4,7) are the vertices of a right triangle.
  9. Triangle ABC
    lalala In a triangle ABC with the side BC of length 2 cm The middle point of AB. Points L and M split AC side into three equal lines. KLM is isosceles triangle with a right angle at the point K. Determine the lengths of the sides AB, AC triangle ABC.
  10. Bisectors
    right_triangle As shown, in △ ABC, ∠C = 90°, AD bisects ∠BAC, DE⊥AB to E, BE = 2, BC = 6. Find the perimeter of triangle △ BDE.
  11. Parametric form
    vzdalenost 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. ..
  12. Medians and sides
    taznice3 Triangle ABC in the plane Oxy; are the coordinates of the points: A = 2.7 B = -4.3 C-6-1 Try calculate lengths of all medians and all sides.
  13. Add vector
    vectors_2 Given that P = (5, 8) and Q = (6, 9), find the component form and magnitude of vector PQ.
  14. Unit vector 2D
    one_1 Determine coordinates of unit vector to vector AB if A[-6; 8], B[-18; 10].
  15. Vector 7
    vectors_sum0_1 Given vector OA(12,16) and vector OB(4,1). Find vector AB and vector |A|.
  16. Isosceles IV
    iso_triangle In an isosceles triangle ABC is |AC| = |BC| = 13 and |AB| = 10. Calculate the radius of the inscribed (r) and described (R) circle.
  17. Theorem prove
    thales_1 We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?