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A spelunker is surveying a cave. She follows a passage 150m

A spelunker is surveying a cave. She follows a passage 150m straight west, then 230m in a
direction 45∘ east of south, and then 280 m at 30∘ east of north. After a fourth unmeasured
displacement, she finds herself back where she started. Determine the magnitude and direction of
the fourth displacement.
Solution:
Second displacement:
Resultant vector π‘Žβƒ— :
π‘Žβƒ— = βƒ—βƒ—βƒ—βƒ—βƒ—
π‘ŽπΈ + βƒ—βƒ—βƒ—βƒ—βƒ—
π‘Žπ‘†
Along the horizontal axis:
π‘ŽπΈ = βˆ’150π‘š + 230m βˆ— cos 45π‘œ = 12.6 π‘š
Along the vertical axis:
π‘Žπ‘† = 230m βˆ— sin 45π‘œ = 162.6 π‘š
Third displacement:
Resultant vector 𝑏⃗⃗ :
𝑏⃗⃗ = βƒ—βƒ—βƒ—βƒ—βƒ—
𝑏𝐸 + βƒ—βƒ—βƒ—βƒ—
𝑏𝑆
Along the horizontal axis:
𝑏𝐸 = π‘ŽπΈ + 280m βˆ— sin 30π‘œ = 152.6 π‘š
Along the vertical axis:
𝑏𝑆 = βˆ’π‘Žπ‘† + 280m βˆ— cos 30π‘œ = 79.8 π‘š
Vector to be found - a vector of the opposite
vector 𝑏⃗⃗:
𝑐⃗ = βˆ’π‘βƒ—βƒ—
Length of the vector 𝑏⃗⃗ , Pythagorean Theorem:
|𝑏| = βˆšπ‘πΈ 2 + 𝑠 2 = √152.6 2 + 79.8 2
= πŸπŸ•πŸ. πŸ•π’Ž
Angle of the vector 𝑐⃗:
𝛼 = arcsin
𝑏𝐸
152.6 π‘š
= arcsin
= πŸ”πŸπ’
𝑏
172.7π‘š
So, the fourth displacement has magnitude 172.7π‘š π‘Žπ‘›π‘‘ π‘‘π‘–π‘Ÿπ‘’π‘π‘‘π‘–π‘œπ‘› π‘Žπ‘‘ 62π‘œ π‘ π‘œπ‘’π‘‘β„Ž π‘œπ‘“ 𝑀𝑒𝑠𝑑
Answer: fourth displacement: 172.7π‘š π‘Žπ‘‘ 62π‘œ π‘ π‘œπ‘’π‘‘β„Ž π‘œπ‘“ 𝑀𝑒𝑠𝑑.

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