SHS2 Additional Mathematics · Semester 2, Week 2
Spatial Sense
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Curriculum details
- Strand
- Geometric Reasoning and Measurement (Strand 2)
- Sub-strand
- Spatial Sense (2.1)
- Content standard
- 2.2.1.CS.2 - Demonstrate knowledge and understanding of spatial sense in relation to related problems. 2.2.1.LO.1 Deduce the equation of a circle and find its centre and radius. 2.2.1.LO.2 Determine the equation of a locus under a given condition.
- Indicator
- 2.2.1.LI.2 - Derive the rule for scalar (dot) product and use it to solve problems relating to angles between two vectors.
- Suggested placement
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Semester 2, Week 2
(Week 22 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
- Source document note
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The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:
- exemplars - p.351: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
- exemplars - p.352: a stacked fraction, matrix, vector or other two-dimensional construct is flattened by the text layer; all visible parts require comparison with the rendered page
- Curriculum reference
- NaCCA curriculum document, p. 351
Exemplars (from the NaCCA curriculum)
Collaborative Learning, Talk for Learning and building on what others. Learning Experience: Learners in collaborative groups derive the scalar (dot) product and solve problems relating to angles between two vectors. Exemplars: Activity 1: Scalar (dot) product Learners in groups deduce the scalar (dot) product for any two vectors 𝑎 and 𝑏 as |𝑎||𝑏|𝑐, as shown in the figure
From the figure, 𝑎 ∙ 𝑏 (read as 𝑎 dot 𝑏) is the product of the magnitude of 𝑎 and 𝑏 and the cosine of the angle between the vectors 𝑎 and 𝑏. Symbolically: 𝑎 ∙ 𝑏 = |𝑎||𝑏|𝑐 Therefore 𝑎 ∙ 𝑏 𝑐 = |𝑎||𝑏| Note: - 𝑎 ∙ 𝑏 is a scalar and not a vector, and - 𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎 Activity 2: Properties of the scalar (dot) product. Learners in groups discuss and establish the properties of the scalar (dot) product. Example 1: If 𝑎 = (𝑝 𝑞 ) and 𝑏 = (𝑟 𝑠 ) then 𝑎 ∙ 𝑏 = 𝑝 + 𝑞 ... ... ... (1) Also 𝑝 + 𝑞 𝑐 = |𝑎||𝑏| Hence 𝑝 + 𝑞 𝜃 = 𝑐 *! £ ¤ |𝑎||𝑏| Putting 𝑝 = 𝑟 and 𝑞 = 𝑠 in equation (1) gives 𝑎 ∙ 𝑎 = 𝑝 ) + 𝑞 ) 𝑝 ) + 𝑞 ) = |𝑎| ) Property 1: The scalar product of a vector with itself equals the square of its magnitude. That is 𝐚 ∙ 𝐚 = 𝐚 𝟐 . Activity: Learners in their groups discuss what happens when two vectors 𝑎 and 𝑏 have the same direction. Property 2: Parallelism property. When two vectors 𝐚 and 𝐛 have the same direction, ã = 𝟎° and 𝐚 ∙ 𝐛 = |𝐚||𝐛|𝐜° then 𝛉 𝑎 ∙ 𝑏 = |𝑎||𝑏| Activity: Learners explore using the formula of the dot product to ascertain what happens if two vectors 𝑎 and 𝑏 are perpendicular. Property 3: If two vectors 𝑎 and 𝑏 are perpendicular, then 𝜃 = 90° and 𝑎 ∙ 𝑏 = |𝑎||𝑏|𝑐90° So that 𝑎 ∙ 𝑏 = 0 Example: Given that 𝑟 = 3𝑖 + 4𝑗 and 𝑡 = 𝑚 − 6𝑗 are perpendicular. Find the value of the constant 𝑚. Thus 𝑟 and 𝑡 are perpendicular, so 𝑟 ∙ 𝑡 = 0. (3 4 ) ∙ (𝑚 − 6 ) = 0 3𝑚 − 24 = 0 𝑚 = 8 Activity 3: Learners in groups collaborate and draw pairs of vectors and measure the angles between them. Learners then discuss how their answers relate to the scalar dot product. Other properties: - Commutativity: The Dot product of two vectors is commutative 𝑎 ∙ 𝑏 = 𝑏. 𝑎 = |𝑎||𝑏| - Multiplication by constant: 𝑝 ∙ 𝑞 = 𝑝|𝑎||𝑏| where p and q are constants, and a and b are vectors - The Dot product follows the distributive law 𝑎 ∙ (𝑏 + 𝑐) = 𝑎. 𝑏 + 𝑏. 𝑐 where 𝑎, 𝑏 and 𝑐 are vectors Example: Find the angle between the vectors 𝑎 = 2𝑖 + 3𝑗 and 𝑏 = 4𝑖 − 𝑗 Solution: 𝑎 ∙ 𝑏 = (2𝑖 + 3𝑗) ∙ (4𝑖 − 𝑗) =8−3 =5 |𝑎| = √4 + 9 = √13 |𝑏| = √16 + 1 = √17 So 5 𝜃 = 𝑐 *! √13 × √17 𝜃 = 𝑐 *! 0.3363 𝜃 = 70.4° The angle between the vectors 2𝑖 + 3𝑗 and 4𝑖 − 𝑗 if 70.4° Teaching and Learning Resources: - SHS curriculum, Mathematical set, ICT apps Assessment (2.2.1.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.