Enrichment, Extension, and Application

Question 1

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Exercises

  1. Triangle \( OAB \) has a right angle at \( O \) and sides \( OA = 8, OB = 6 \). \( M \) is the midpoint of the side \( AB \). The segment \( OC = 12 \) is perpendicular to the plane of the triangle. Find the measure of \( \angle{CMB} \), to the nearest tenth of a degree.
  2. Find all unit vectors perpendicular to \( (1, 2, 3) \) that make equal angles with the unit vectors \( \hat{i} \) and \( \hat{j} \).
  3. Image described in question. Let \( \vec{u}, \vec{v} \) be two unit vectors. Let \( \alpha \) be the angle between \( \vec{u} \) and \( \hat{i} \), and \( \beta \) the angle between \( \vec{v} \) and \( \hat{i} \), as shown in the diagram.
    1. Express \( \vec{u} \) in terms of \( \vec{i}\), \(\vec{j} \), and \( \alpha \). Express \( \vec{v} \) in terms of \( \vec{i}\), \(\vec{j} \), and \( \beta \).
    2. Use the dot product of the vectors \( \vec{u} \) and \( \vec{v} \) to find a formula for \( \cos\left( \alpha - \beta \right) \).
  4. Diagram described in question. In the diagram, \( ABCD \) is a square of side length \( 24 \). \( M \) is the midpoint of \( BC \) and \( N \) is situated on \( DC \), dividing \( DC \) in the ratio \( 1 : 2 \). \( AM \) and \( BN \) intersect at point \( P \). Since \( A, P, M \) are collinear, \( \overrightarrow{AP} = m\overrightarrow{AM} \) for some constant \( m \); similarly, since \( B, P, N \) are collinear, \( \overrightarrow{BP} = n\overrightarrow{BN} \) for some constant \( n \).
    1. Find the constants \( m \) and \( n \). Hint: Choose a coordinate system with \( A \) at the origin and \( AB \) as the \( x \)-axis. Then, write an equation connecting \( \overrightarrow{AP} \) and \( \overrightarrow{BP} \).
    2. Find the area of \( \triangle{PBM} \).
  5. Consider the vectors \( \vec{a}, \vec{b} \), and \( \vec{c} \). If \( \vec{u} = \left( \vec{a} \cdot \vec{b} \right) \vec{c} - \left( \vec{a} \cdot \vec{c} \right) \cdot \vec{b} \), prove that \( \vec{a} \) is perpendicular to \( \vec{u} \).
  6. Triangle \( OAB \) is inscribed in a circle with centre at \( C \). Points \( O~(0, 0)\), \(A~(2a, 2b) \), and \( B~(2c, 0) \) are given.
    1. Express the coordinates of \( C \) in terms of \( a\), \(b \), and \( c \).
    2. Find a point \( H \) such that \( \overrightarrow{CH} = \overrightarrow{CO} + \overrightarrow{CA} + \overrightarrow{CB} \).
    3. Prove that all three altitudes of \( \triangle{ABC} \) pass through \( H \) by showing that \( \overrightarrow{OH} \perp \overrightarrow{AB}\), \(\overrightarrow{BH} \perp \overrightarrow{OA} \), and \( \overrightarrow{AH} \perp \overrightarrow{OB} \).
  7. Vectors \( \vec{u}, \vec{v}, \vec{w} \) are perpendicular to each other and \( \left\lvert \vec{u} \right\rvert = 1, \left\lvert \vec{v} \right\rvert = 3, \left\lvert \vec{w} \right\rvert = 4 \). Find the magnitude of the vector \( \left( \vec{u} \times \vec{v} \right) + \left( \vec{v} \times \vec{w} \right) + \left( \vec{w} \times \vec{u} \right) \).
  8. Vectors \( \vec{u} \) and \( \vec{v} \) are perpendicular and \( \left\lvert \vec{u} \right\rvert = \sqrt{3} \left\lvert \vec{v} \right\rvert \). Find all vectors \( \vec{x} \) which satisfy \( \vec{u} \times \vec{x} = \vec{x} \times \vec{v} \).
  9. Let \( \vec{u} = (-1, 2, 0)\), \(\vec{v} = (-1, 0, 1) \), and \( \vec{w} = (0, 1, 1) \).
    1. Find \( \vec{z} = \left( \vec{u} \times \vec{v} \right) \times \left( \vec{u} \times \vec{w} \right) \), and show that \( \vec{z} \) is collinear with \( \vec{u} \).
    2. Explain why, for any three non-coplanar vectors \( \vec{u}\), \(\vec{v} \), and \( \vec{w} \), the vector \( \vec{z} = \left( \vec{u} \times \vec{v} \right) \times \left( \vec{u} \times \vec{w} \right) \) is collinear with \( \vec{u} \).
  10. Let \( \vec{u} = (1, 2, 3), \vec{v} = (-3, 1, 2) \).
    1. If \( \vec{w} = (-1, 5, 8) \), calculate \( \left( \vec{u} \times \vec{v} \right) \cdot \vec{w} \) and explain why \( \vec{u}\), \(\vec{v} \), and \( \vec{w} \) are coplanar. Express \( \vec{w} \) as a linear combination of the vectors \( \vec{u} \) and \( \vec{v} \).
    2. If \( \vec{z} = (-1, 2, 0) \), show that \( \vec{u}, \vec{v}, \vec{z} \) are not coplanar and hence form a basis for the space. Express \( \vec{x} = (2, 3, 11) \) as a linear combination of the vectors.
  11. Tetrahedron described in question
    1. In the regular tetrahedron \( ABCD \), \( M \) is the midpoint of \( BC \) and \( AH \) is the altitude of triangle \( AMD \). Prove that \( AH \) is perpendicular to the plane containing \( \triangle{BCD} \).
    2. If the sides of tetrahedron \( ABCD \) are length \( 1 \), and \( B \) is the origin of a coordinate system, with \( BC \) lying on the \( x \)-axis, and \( \triangle{BCD} \) in the first quadrant, write \( \overrightarrow{BA}\), \(\overrightarrow{BC} \), and \( \overrightarrow{BD} \) as algebraic vectors.
    3. Calculate the volume of the tetrahedron described in b).
    1. Prove that if \( \overrightarrow{PA_1} + \overrightarrow{PA_2} + \cdots + \overrightarrow{PA_n} = \vec{0} \), then \( \overrightarrow{OP} = \dfrac{\overrightarrow{OA_1} + \overrightarrow{OA_2} + \cdots + \overrightarrow{OA_n}}{n} \).
    2. Tetrahedron with vertices labelled H_1,H_2,H_3,H_4; C is a point equidistant from every vertex

      Tetrahedral structures like the one in problem 9 can appear in nature as the methane molecule \( \text{CH}_4 \), where four hydrogen atoms surround a central carbon atom with tetrahedral symmetry.

      In the following diagram, we have \( \overrightarrow{CH_1} + \overrightarrow{CH_2} + \overrightarrow{CH_3} + \overrightarrow{CH_4} = \vec{0} \) and \( \left\lvert \overrightarrow{CH_1} \right\rvert = \left\lvert \overrightarrow{CH_2} \right\rvert = \left\lvert \overrightarrow{CH_3} \right\rvert = \left\lvert \overrightarrow{CH_4} \right\rvert \). The tetrahedral bond angle is the angle formed between any two bonds; for instance, the angle \( H_1CH_2 \). Calculate this angle.

      Note: \( \text{CH}_4 \) is a chemical formula describing the composition of the entire methane molecule; \( CH_4 \) is the line segment between the points \( C \) and \( H_4 \) in the diagram, representing the bond between the carbon atom and that specific hydrogen atom.

    1. Prove the theorem of Apollonius, which states:
      In \( \triangle{ABC} \), if \( M \) is the midpoint of \( BC \), then \( AB^2 + AC^2 = 2AM^2 + 2MC^2 \)
      Hint: Denote \( \overrightarrow{AB} = \vec{b}\), \(\overrightarrow{AC} = \vec{c} \), and use the dot product.
    2. In \( \triangle{ABC} \), \( AB = 8\), \(AC = 11 \), and \( BC = 9 \). Find the length of the median \( AM \).
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