2020 Calculus II (Math 162-24) HW1 | Korea University
1. Find the distance from (4,-2,6) to the z-axis.
𝐷= √𝑥 2 + 𝑦 2 to the z-axis
= √42 + (−2)2
= √16 + 4 = √20 = 2√5
2. Show that the equation represents a sphere, and find its center and radius.
𝟑𝒙𝟐 + 𝟑𝒚𝟐 − 𝟔𝒚 + 𝟑𝒛𝟐 − 𝟏𝟐𝒛 = 𝟏𝟎
3x 2 + 3y 2 − 6y + 3z 2 − 12z = 10
3𝑥 2 + 3(𝑦 2 − 2𝑦 + 1) + 3(𝑧 2 − 4𝑧 + 4) = 10 + 3 + 12
3𝑥 2 + 3(𝑦 − 1)2 + 3(𝑧 − 2)2 = 25
25
𝑥 2 + (𝑦 − 1)2 + (𝑧 − 2)2 =
3
5
𝑐𝑒𝑛𝑡𝑒𝑟 (0,1,2), 𝑟𝑎𝑑𝑖𝑢𝑠 =
√3
3. Find the vector that has the same direction as <6,2,-3>but has length 4.
| < 6,2, −3 > | = √62 + 22 + (−3)2 = √36 + 4 + 9 = √49 = 7
1
𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑖𝑠 𝑢
⃑ = < 6,2, −3 >
7
𝐴 𝑣𝑒𝑐𝑡𝑜𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑏𝑢𝑡 𝑤𝑖𝑡ℎ 𝑙𝑒𝑛𝑔𝑡ℎ 4 𝑖𝑠
1 4
𝑣=4 ∙ < 6,2, −3 > = < 6,2, −3 >
7 7
4. Find the unit vectors that are parallel to the tangent line to the parabola 𝒚 = 𝒙𝟐 at the
point (2,4)
𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑡𝑜 𝑦 = 𝑥 2 𝑎𝑡 (2,4) 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑢𝑛𝑑 𝑖𝑛
𝑑𝑦
| = 2𝑥|𝑥=2 = 4
𝑑𝑥 𝑥=2
𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 = 𝑖 + 4𝑗
𝑣 1
𝑠𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝑣 𝑖𝑠 𝑢
⃑ = = (𝑖 + 4𝑗)
|𝑣| √1 + 16
1
∴ ±𝑣 = ± (𝑖 + 4𝑗)
√17
5. Find a unit vector that is orthogonal to both 𝒊 + 𝒋 and 𝒊 + 𝒌
⃑
𝑙𝑒𝑡 𝑎 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 = 𝑣1 𝑖 + 𝑣2 𝑗 + 𝑣3 𝑘 𝑏𝑒 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙 𝑡𝑜 𝑏𝑜𝑡ℎ 𝑖 + 𝑗 𝑎𝑛𝑑 𝑖 + 𝑘⃑
1. Find the distance from (4,-2,6) to the z-axis.
𝐷= √𝑥 2 + 𝑦 2 to the z-axis
= √42 + (−2)2
= √16 + 4 = √20 = 2√5
2. Show that the equation represents a sphere, and find its center and radius.
𝟑𝒙𝟐 + 𝟑𝒚𝟐 − 𝟔𝒚 + 𝟑𝒛𝟐 − 𝟏𝟐𝒛 = 𝟏𝟎
3x 2 + 3y 2 − 6y + 3z 2 − 12z = 10
3𝑥 2 + 3(𝑦 2 − 2𝑦 + 1) + 3(𝑧 2 − 4𝑧 + 4) = 10 + 3 + 12
3𝑥 2 + 3(𝑦 − 1)2 + 3(𝑧 − 2)2 = 25
25
𝑥 2 + (𝑦 − 1)2 + (𝑧 − 2)2 =
3
5
𝑐𝑒𝑛𝑡𝑒𝑟 (0,1,2), 𝑟𝑎𝑑𝑖𝑢𝑠 =
√3
3. Find the vector that has the same direction as <6,2,-3>but has length 4.
| < 6,2, −3 > | = √62 + 22 + (−3)2 = √36 + 4 + 9 = √49 = 7
1
𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑖𝑠 𝑢
⃑ = < 6,2, −3 >
7
𝐴 𝑣𝑒𝑐𝑡𝑜𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑏𝑢𝑡 𝑤𝑖𝑡ℎ 𝑙𝑒𝑛𝑔𝑡ℎ 4 𝑖𝑠
1 4
𝑣=4 ∙ < 6,2, −3 > = < 6,2, −3 >
7 7
4. Find the unit vectors that are parallel to the tangent line to the parabola 𝒚 = 𝒙𝟐 at the
point (2,4)
𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑡𝑜 𝑦 = 𝑥 2 𝑎𝑡 (2,4) 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑢𝑛𝑑 𝑖𝑛
𝑑𝑦
| = 2𝑥|𝑥=2 = 4
𝑑𝑥 𝑥=2
𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 = 𝑖 + 4𝑗
𝑣 1
𝑠𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝑣 𝑖𝑠 𝑢
⃑ = = (𝑖 + 4𝑗)
|𝑣| √1 + 16
1
∴ ±𝑣 = ± (𝑖 + 4𝑗)
√17
5. Find a unit vector that is orthogonal to both 𝒊 + 𝒋 and 𝒊 + 𝒌
⃑
𝑙𝑒𝑡 𝑎 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑣 = 𝑣1 𝑖 + 𝑣2 𝑗 + 𝑣3 𝑘 𝑏𝑒 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙 𝑡𝑜 𝑏𝑜𝑡ℎ 𝑖 + 𝑗 𝑎𝑛𝑑 𝑖 + 𝑘⃑
