Math Problem Statement
2 The diagram shows a cuboid ABCDEFGH with horizontal base ADEH Diagram NOT accurately drawn 20 cm 12 cm 8 cm A B C M H D E G F P AB = 8 cm AD = 12 cm DE = 20 cm M is the midpoint of the base ADEH and P is the midpoint of the edge CF Work out the size of angle BMP Give your answer correct to one decimal place
Solution
To find the size of angle in the cuboid, we will:
-
Identify key points and coordinates:
- Place the cuboid in a 3D coordinate system with .
- Define the key points based on the dimensions:
- , , , ,
- , , , .
- , the midpoint of the base :
- , the midpoint of edge :
-
Calculate vectors and :
- .
- .
-
Find the dot product of and :
- Dot product:
-
Find the magnitudes of and :
- Magnitude of :
- Magnitude of :
-
Calculate :
- Using the formula for the angle between two vectors:
- Substitute the values:
-
Find :
- Use :
Final Answer:
The size of angle is 50.1°, correct to one decimal place.
Would you like further clarification on any step, or assistance with another problem? 😊
Relative Questions:
- What is the formula for the dot product of two vectors in 3D space?
- How do you calculate the midpoint of a line segment in a 3D coordinate system?
- What is the significance of the angle between two vectors in geometry?
- How do you simplify square roots involving large numbers in magnitude calculations?
- Why is it important to verify units and scales in a problem involving 3D geometry?
Tip:
When working with vectors in 3D, always double-check your coordinates and calculations for midpoint, as errors propagate quickly!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
3D Geometry
Vectors
Dot Product
Midpoint
Angle between Vectors
Formulas
Dot product of vectors: \vec{A} \cdot \vec{B} = |A| |B| \cos(\theta)
Magnitude of a vector: |\vec{A}| = \sqrt{x^2 + y^2 + z^2}
Midpoint formula in 3D: M = (x_1 + x_2)/2, (y_1 + y_2)/2, (z_1 + z_2)/2
Theorems
Dot Product Theorem
Cosine Rule for Angle Between Vectors
Suitable Grade Level
Grades 10-12