Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Start Practicing

Worksheet: Geometric and Physical Applications of Vectors

Q1:

Given that ⃑ 𝑣 = 1 7 ⃑ 𝑖 𝐴 and ⃑ 𝑣 = 8 ⃑ 𝑖 𝐡 , find ⃑ 𝑣 𝐡 𝐴 .

  • A 2 5 ⃑ 𝑖
  • B 9 ⃑ 𝑖
  • C βˆ’ 2 5 ⃑ 𝑖
  • D βˆ’ 9 ⃑ 𝑖

Q2:

The forces ⃑ 𝐹 = ο€Ί βˆ’ 4 ⃑ 𝑖 + 6 ⃑ 𝑗  1 N , ⃑ 𝐹 = ο€Ί βˆ’ 9 ⃑ 𝑖 + 4 ⃑ 𝑗  2 N , and ⃑ 𝐹 = ο€Ί βˆ’ 4 ⃑ 𝑖 βˆ’ 3 ⃑ 𝑗  3 N are acting on a particle, where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Determine the magnitude the forces’ resultant, 𝑅 , and its direction πœƒ to the nearest minute.

  • A 𝑅 = 1 3 √ 2 N , πœƒ = 1 1 2 2 3 β€² ∘
  • B 𝑅 = 3 3 8 N , πœƒ = 1 1 2 2 3 β€² ∘
  • C 𝑅 = 2 √ 6 N , πœƒ = 1 5 7 3 7 β€² ∘
  • D 𝑅 = 1 3 √ 2 N , πœƒ = 1 5 7 3 7 β€² ∘

Q3:

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

Q4:

If ⃑ 𝑣 = βˆ’ 7 6 ⃑ 𝑖 𝐴 𝐡 and ⃑ 𝑣 = βˆ’ 5 ⃑ 𝑖 𝐴 , find ⃑ 𝑣 𝐡 .

  • A βˆ’ 7 1 ⃑ 𝑖
  • B 8 1 ⃑ 𝑖
  • C βˆ’ 8 1 ⃑ 𝑖
  • D 7 1 ⃑ 𝑖

Q5:

A book moves on a horizontal plane after being pushed by a force of 12 N parallel to the plane. The friction force between the book and the plane is 5 N. Find the resultant of these forces, and express it in terms of ⃑ 𝑒 , the unit vector in the direction of the book’s motion.

  • A ο€Ή 1 2 ⃑ 𝑒  N
  • B ο€Ή 1 7 ⃑ 𝑒  N
  • C ο€Ή 5 ⃑ 𝑒  N
  • D ο€Ή 7 ⃑ 𝑒  N

Q6:

The forces ⃑ 𝐹 = βˆ’ 1 0 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗 1 , ⃑ 𝐹 = π‘Ž ⃑ 𝑖 βˆ’ ⃑ 𝑗 2 , and ⃑ 𝐹 = 5 ⃑ 𝑖 + ( 𝑏 βˆ’ 1 0 ) ⃑ 𝑗 3 act on a particle, where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Given that the forces’ resultant ⃑ 𝑅 = βˆ’ 1 3 ⃑ 𝑖 βˆ’ 3 ⃑ 𝑗 , determine the values of π‘Ž and 𝑏 .

  • A π‘Ž = 2 , 𝑏 = 1 5
  • B π‘Ž = βˆ’ 8 , 𝑏 = βˆ’ 5
  • C π‘Ž = βˆ’ 8 , 𝑏 = 5
  • D π‘Ž = βˆ’ 8 , 𝑏 = 1 5
  • E π‘Ž = βˆ’ 2 8 , 𝑏 = βˆ’ 5

Q7:

The resultant of forces ⃑ 𝐹 = ο€Ί βˆ’ 4 ⃑ 𝑖 + 2 ⃑ 𝑗  1 N , ⃑ 𝐹 = ο€Ί 5 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗  2 N , and ⃑ 𝐹 = ο€Ί 2 ⃑ 𝑖 + 9 ⃑ 𝑗  3 N , makes an angle πœƒ with the positive π‘₯ -axis. Determine 𝑅 , the magnitude of the resultant, and the value of t a n πœƒ .

  • A 𝑅 = 7 N , t a n πœƒ = 4 3
  • B 𝑅 = 5 N , t a n πœƒ = 3 4
  • C 𝑅 = 7 N , t a n πœƒ = 3 4
  • D 𝑅 = 5 N , t a n πœƒ = 4 3
  • E 𝑅 = 5 N , t a n πœƒ = βˆ’ 4 3

Q8:

Given that ⃑ 𝐹 = 8 ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗 1 , ⃑ 𝐹 = βˆ’ 1 5 ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗 2 , and their resultant ⃑ 𝑅 = βˆ’ π‘Ž ⃑ 𝑖 βˆ’ 𝑏 ⃑ 𝑗 , determine the values of π‘Ž and 𝑏 .

  • A π‘Ž = 7 , 𝑏 = 0
  • B π‘Ž = βˆ’ 2 3 , 𝑏 = 1 0
  • C π‘Ž = βˆ’ 7 , 𝑏 = 1 0
  • D π‘Ž = 7 , 𝑏 = 1 0
  • E π‘Ž = 7 , 𝑏 = βˆ’ 1 0

Q9:

All the sides of rhombus 𝑂 𝐡 𝐢 𝐴 are of length 5. Suppose that s i n ∠ 𝐴 𝑂 𝐡 = 3 4 and that 𝐴 𝐡 > 𝑂 𝐢 . Use vector multiplication to find the lengths of the two diagonals.

  • A 𝑂 𝐢 = 4 . 1 1 , 𝐴 𝐡 = 3 . 2 7
  • B 𝑂 𝐢 = 1 . 8 4 , 𝐴 𝐡 = 4 . 0 8
  • C 𝑂 𝐢 = 1 6 . 9 3 , 𝐴 𝐡 = 2 6 . 5 4
  • D 𝑂 𝐢 = 4 . 1 1 , 𝐴 𝐡 = 9 . 1 1

Q10:

𝐴 𝐡 𝐢 𝐷 is a square, in which the coordinates of the points 𝐴 , 𝐡 , and 𝐢 are ( 1 , βˆ’ 8 ) , ( 3 , βˆ’ 1 0 ) , and ( 5 , βˆ’ 8 ) . Use vectors to determine the coordinates of the point 𝐷 and the area of the square.

  • A 𝐷 ( 9 , βˆ’ 2 6 ) , area = 1 6
  • B 𝐷 ( 1 , 1 0 ) , area = 3 4 0
  • C 𝐷 ( 7 , βˆ’ 1 0 ) , area = 8
  • D 𝐷 ( 3 , βˆ’ 6 ) , area = 8

Q11:

Given that ⃑ 𝐴 = ( βˆ’ 2 , 7 ) and ⃑ 𝐡 = ( 3 , βˆ’ 8 ) , determine the area of the parallelogram whose adjacent sides are represented by ⃑ 𝐴 and ⃑ 𝐡 .

Q12:

Trapezium 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( 4 , 1 4 ) , 𝐡 ( 4 , βˆ’ 4 ) , 𝐢 ( βˆ’ 1 2 , βˆ’ 4 ) , and 𝐷 ( βˆ’ 1 2 , 9 ) . Given that οƒ  𝐴 𝐡 β«½ οƒ  𝐷 𝐢 and οƒ  𝐴 𝐡 βŸ‚ οƒŸ 𝐢 𝐡 , find the area of that trapezium.

Q13:

Given a trapezium 𝐴 𝐡 𝐢 𝐷 , in which 𝐴 𝐷 βˆ₯ 𝐡 𝐢 and 𝐴 𝐷 𝐡 𝐢 = 7 , find the value of π‘˜ such that οƒ  𝐴 𝐢 + οƒ  𝐡 𝐷 = π‘˜ οƒ  𝐴 𝐷 .

  • A 1 7
  • B8
  • C 1 5 7
  • D 8 7

Q14:

𝐴 𝐡 𝐢 𝐷 is a rectangle, in which the coordinates of the points 𝐴 , 𝐡 , and 𝐢 are ( βˆ’ 1 8 , βˆ’ 2 ) , ( βˆ’ 1 8 , βˆ’ 3 ) , and ( βˆ’ 8 , π‘˜ ) , respectively. Use vectors to find the value of π‘˜ and the coordinates of point 𝐷 .

  • A π‘˜ = βˆ’ 2 , 𝐷 ( βˆ’ 2 8 , βˆ’ 2 )
  • B π‘˜ = βˆ’ 2 , 𝐷 ( βˆ’ 8 , βˆ’ 2 )
  • C π‘˜ = βˆ’ 1 , 𝐷 ( βˆ’ 2 8 , βˆ’ 2 )
  • D π‘˜ = βˆ’ 3 , 𝐷 ( βˆ’ 8 , βˆ’ 2 )
  • E π‘˜ = βˆ’ 1 , 𝐷 ( βˆ’ 8 , βˆ’ 3 )

Q15:

Given the information in the diagram below, find the value of 𝑛 such that οƒ  𝐴 𝐷 + οƒ  𝐷 𝐸 = 𝑛 οƒ  𝐴 𝐢 .

  • A 6 7
  • B βˆ’ 1 2
  • C βˆ’ 6 7
  • D 1 2

Q16:

Given a triangle 𝐴 𝐡 𝐢 , in which 𝐴 𝐡 = 7 c m , 𝐡 𝐢 = 5 6 c m , and π‘š ∠ 𝐴 𝐡 𝐢 = 1 2 0 ∘ , use vectors to determine the length of 𝐴 𝐢 .

  • A 1 1 √ 7 cm
  • B 7 √ 5 7 cm
  • C 2 √ 7 9 8 cm
  • D 7 √ 7 3 cm

Q17:

In the given figure, βƒ–     βƒ— 𝐴 𝐡 and βƒ–     βƒ— 𝐢 𝐷 are parallel lines; however, βƒ–      βƒ— 𝑋 π‘Œ is NOT parallel to either βƒ–     βƒ— 𝐴 𝐡 or βƒ–     βƒ— 𝐢 𝐷 . Given that 𝐸 ∈ 𝐴 𝐡 , 𝐹 ∈ 𝐢 𝐷 , and 𝑍 ∈ 𝑋 π‘Œ , determine whether  π‘Œ 𝑍 and  𝑋 𝑍 are in the same, opposite, or different directions.

  • Adifferent
  • Bsame
  • Copposite