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Lesson: Geometrical Interpretation of the Vector Product

Worksheet • 17 Questions

Q1:

Find the area of a triangle 𝐴 𝐡 𝐢 , where 𝐴 ( βˆ’ 8 , βˆ’ 9 ) , 𝐡 ( βˆ’ 7 , βˆ’ 8 ) , and 𝐢 ( 9 , βˆ’ 2 ) .

Q2:

Find the area of a triangle 𝐴 𝐡 𝐢 , where 𝐴 ( 0 , 5 ) , 𝐡 ( 7 , βˆ’ 9 ) , and 𝐢 ( βˆ’ 1 , 0 ) .

Q3:

Find the area of a triangle 𝐴 𝐡 𝐢 , where 𝐴 ( 9 , βˆ’ 3 ) , 𝐡 ( βˆ’ 3 , βˆ’ 2 ) , and 𝐢 ( βˆ’ 9 , 2 ) .

Q4:

Find the area of a triangle 𝐴 𝐡 𝐢 , where 𝐴 ( 1 , 4 ) , 𝐡 ( βˆ’ 7 , 6 ) , and 𝐢 ( βˆ’ 9 , βˆ’ 5 ) .

Q5:

Assuming that ( ⃑ 𝑖 , ⃑ 𝑗 , ⃑ π‘˜ ) form a right-hand system, ⃑ 𝐴 = 1 6 ⃑ 𝑖 + 4 ⃑ 𝑗 , ⃑ 𝐡 = 1 9 ⃑ 𝑖 + 8 ⃑ 𝑗 , and ⃑ 𝐴 and ⃑ 𝐡 form two adjacent sides of a triangle, find the vector product of ⃑ 𝐴 into ⃑ 𝐡 and the area of the triangle drawn.

  • A ⃑ 𝐴 Γ— ⃑ 𝐡 = 5 2 ⃑ π‘˜ , area = 2 6 square units
  • B ⃑ 𝐴 Γ— ⃑ 𝐡 = 3 3 6 ⃑ π‘˜ , area = 1 6 8 square units
  • C ⃑ 𝐴 Γ— ⃑ 𝐡 = 2 0 4 ⃑ π‘˜ , area = 1 0 2 square units
  • D ⃑ 𝐴 Γ— ⃑ 𝐡 = 2 7 2 ⃑ π‘˜ , area = 1 3 6 square units
  • E ⃑ 𝐴 Γ— ⃑ 𝐡 = βˆ’ 8 8 ⃑ π‘˜ , area = 4 4 square units

Q6:

𝐴 𝐡 𝐢 𝐷 is a rhombus, in which the coordinates of the points 𝐴 and 𝐡 are ( 5 , βˆ’ 9 ) and ( βˆ’ 1 0 , 1 2 ) , respectively. Use vectors to determine its perimeter.

  • A 1 2 √ 7 4 length units
  • B 3 √ 7 4 length units
  • C 6 √ 7 4 length units
  • D666 length units

Q7:

If 𝐴 𝐡 𝐢 is a triangle of area 248.5 cm2, find the value of β€– β€– οƒ  𝐡 𝐴 Γ— οƒ  𝐴 𝐢 β€– β€– .

Q8:

Rhombus 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( βˆ’ 4 , 6 ) , 𝐡 ( 9 , 2 ) , 𝐢 ( βˆ’ 2 , 1 0 ) , and 𝐷 ( βˆ’ 1 5 , 1 4 ) . Use vectors to determine its area.

Q9:

Rhombus 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( 7 , βˆ’ 6 ) , 𝐡 ( 0 , βˆ’ 2 ) , 𝐢 ( βˆ’ 1 , 6 ) , and 𝐷 ( 6 , 2 ) . Use vectors to determine its area.

Q10:

Rhombus 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( 5 , 6 ) , 𝐡 ( βˆ’ 5 , 1 ) , 𝐢 ( βˆ’ 3 , 1 2 ) , and 𝐷 ( 7 , 1 7 ) . Use vectors to determine its area.

Q11:

Rhombus 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( 6 , 1 0 ) , 𝐡 ( 2 , βˆ’ 7 ) , 𝐢 ( βˆ’ 5 , 9 ) , and 𝐷 ( βˆ’ 1 , 2 6 ) . Use vectors to determine its area.

Q12:

Given that 𝐷 = ( 0 , βˆ’ 2 , βˆ’ 8 ) , 𝐸 = ( 6 , 4 , 6 ) , and 𝐹 = ( βˆ’ 4 , βˆ’ 9 , βˆ’ 2 ) , determine the area of the triangle 𝐷 𝐸 𝐹 approximated to the nearest hundredth.

Q13:

Given that 𝐷 = ( βˆ’ 2 , 0 , βˆ’ 2 ) , 𝐸 = ( βˆ’ 3 , βˆ’ 9 , 5 ) , and 𝐹 = ( βˆ’ 5 , βˆ’ 1 , βˆ’ 6 ) , determine the area of the triangle 𝐷 𝐸 𝐹 approximated to the nearest hundredth.

Q14:

Triangle 𝐴 𝐡 𝐢 has vertices 𝐴 ( 5 , βˆ’ 4 ) , 𝐡 ( βˆ’ 1 , βˆ’ 5 ) , and 𝐢 ( βˆ’ 3 , 2 ) . Use vectors to determine its area.

Q15:

Suppose that ⃑ 𝐴 = ( 1 , 1 , 3 ) and ⃑ 𝐡 = ( 4 , 8 , βˆ’ 8 ) fix two sides of a triangle. What is the area of this triangle, to the nearest hundredth?

Q16:

Suppose that ⃑ 𝐴 = ( βˆ’ 5 , 4 , 5 ) and ⃑ 𝐡 = ( βˆ’ 1 , 8 , 2 ) fix two sides of a triangle. What is the area of this triangle, to the nearest hundredth?

Q17:

Suppose that ⃑ 𝐴 = ( βˆ’ 2 , βˆ’ 8 , 8 ) and ⃑ 𝐡 = ( 3 , βˆ’ 3 , βˆ’ 3 ) fix two sides of a triangle. What is the area of this triangle, to the nearest hundredth?

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