Step 1: Find the Component Form of the Vector The component form of a vector is found by considering the displacement along each axis from the starting point to the ending point. In this case, the starting point is the origin (0,0,0), and the ending point is the opposite corner of the prism. Along the x-axis, the displacement is equal to the width of the prism, which is 8 cm. Along the y-axis, the displacement is equal to the depth of the prism, which is 5 cm. Along the z-axis, the displacement is equal to the height of the prism, which is 12 cm. So, the vector 𝑝 ⃗ p in component form is ⟨ 8 , 5 , 12 ⟩ ⟨ 8 , 5 , 12 ⟩ . Explanation: Determine the vector's components by measuring the change along each axis from the origin to the prism's far corner, resulting in 𝑝 ⃗ = ⟨ 8 , 5 , 12 ⟩ p = ⟨ 8 , 5 , 12 ⟩ . Step 2: Calculate the Magnitude of the Vector The magnitude of a vector in three-dimensional space is the distance from the starting point to the ending point and can be calculated using the Pythagorean theorem extended to three dimensions. This is done by taking the square root of the sum of the squares of the components. For the vector 𝑝 ⃗ = ⟨ 8 , 5 , 12 ⟩ p = ⟨ 8 , 5 , 12 ⟩ , the magnitude is calculated as follows: ∣ 𝑝 ⃗ ∣ = 𝑥 2 + 𝑦 2 + 𝑧 2 ∣ p ∣ = x 2 + y 2 + z 2 ∣ 𝑝 ⃗ ∣ = 8 2 + 5 2 + 1 2 2 ∣ p ∣ = 8 2 + 5 2 + 1 2 2 ∣ 𝑝 ⃗ ∣ = 64 + 25 + 144 ∣ p ∣ = 64 + 25 + 144 ∣ 𝑝 ⃗ ∣ = 233 ∣ p ∣ = 233 = = 15.26cm Explanation: Calculate the vector's magnitude by using the formula ∣ 𝑝 ⃗ ∣ = 𝑥 2 + 𝑦 2 + 𝑧 2 ∣ p ∣ = x 2 + y 2 + z 2 , yielding 15.26 cm. Final Answer: a. Give the dashed, purple vector p in component form: < number, number, number >. Answer: The dashed, purple vector 𝑝 ⃗ p in component form is 𝑝 ⃗ = < 8 , 12 , 5 > p =< 8 , 12 , 5 > . b. What is the magnitude of this vector? Give your answer as a decimal number with units Answer: The magnitude of the vector 𝑝 ⃗ p is 15.26 cm

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