Additional Exercises for the 3rd Exam in Math 54 + = + if and are Is it possible to have non-zero vectors? 2. Find vectors that satisfy the stated conditions. a. Oppositely directed to = ⌊3, −4âŒŞ and half the length of b. Length √17 and same direction as = ⌊7,0, −6âŒŞ c. Same direction as = −2 Ě‚ + 3 Ě‚ and three times the length of d. Length 2 and oppositely directed to = −3 Ě‚ + 4 Ě‚ + e. Length 3 and same direction as + , where = Ě‚ + 2 Ě‚ − 3 , and = ⌊2,2, −4âŒŞ 3. In each part, determine whether and make an acute angle, an obtuse angle or are orthogonal. a. = 7 Ě‚ + 3 Ě‚ + 5 , = −8 Ě‚ + 4 Ě‚ + 2 b. = 6 Ě‚ + Ě‚ + 3 , = 4 Ě‚ − 6 c. = ⌊1,1,1âŒŞ, = ⌊−1,0,0âŒŞ d. = ⌊4,1,6âŒŞ, = ⌊−3,0,2âŒŞ 4. Find the cross product of and and check that it is orthogonal to both and . a. = ⌊1,2, −3âŒŞ, = ⌊−4,1,2âŒŞ b. = 3 Ě‚ + 2 Ě‚ = , = − Ě‚ − 3 Ě‚ + c. = 4 Ě‚ + , = 2 Ě‚ − Ě‚ 5. Find the area of the parallelogram that has = Ě‚ − Ě‚ + 2 and = 3 Ě‚ + as adjacent sides. 6. Prove Lagrange’s identity for vectors in 3-space that ‖ Ă— ‖ = ‖ ‖ ‖ ‖ − ! â‹… # . 7. Given lines $% : ' = 1 + 2(, ) = 2 − (, * = 4 − 2( and $ : ' = 9 + (, ) = 5 + 3(, * = −4 − (. a. Show that $% and $ intersect at the point !7, −1,2#. b. Find parametric equations for the line that is perpendicular to $% and $ and passes through their point of intersection. 8. Find an equation of the plane that passes through the given points. a. !−2,1,1#, !0,2,3#, and !1,0, −1# b. !3,2,1#, !2,1, −1#, and !−1,3,2# 9. Determine whether the planes are parallel, perpendicular, or neither. a. 2' − 8) − 6* − 2 = 0 −' + 4) + 3* − 5 = 0 b. 3' − 2) + * = 1, 4' + 5) − 2* = 4 c. ' − 2 + 3* − 2 = 0, 2' + * = 1 10. Determine whether the line and plane are parallel, perpendicular or neither. a. ' = 4 + 2(, ) = −(, * = −1 − 4(; 3' + 2) + * − 7 = 0 b. ' = (, ) = 2(, * = 3(; ' − ) + 2* = 5 c. ' = −1 + 2(, ) = 4 + (, * = 1 − 3(; '+)+*=1 11. Determine whether the line and plane intersect; if so, find the coordinates of the intersection. a. ' = (, ) = (, * = ( 3' − 2) + * − 5 = 0 b. ' = 2 − (, ) = 3 + (, * = ( 2' + ) + * = 1 c. ' = 1 + (, ) = −1 + 3(, * = 2 + 4( ' − ) + 4* = 7
1.
12. Find an equation of the plane that satisfies the stated conditions. a. The plane through the origin that is parallel to the plane 4' − 2) + 7* + 12 = 0. b. The plane that contains the line ' = −2 + 3(, ) = 4 + 2(, * = 3 − ( and is perpendicular to the plane ' − 2) + * = 5. c. The plane through the point !−1,4,2# that contains the line of intersection of the planes 4' − ) + * − 2 = 0 and 2' + ) − 2* − 3 = 0. 13. Find parametric equations of the line of intersection of the planes. a. −2' + 3) + 7* + 2 = 0, ' + 2) − 3* + 5 = 0 b. 3' − 5) + 2* = 0, * = 0 14. Find the distance between the point and the plane. a. !1, −2,3#, 2' − 2) + * = 4 b. !0,1,5#, 3' + 6) − 2* − 5 = 0 15. Find an equation of the sphere with center at !2,1, −3# that is tangent to the plane ' − 3) + 2* = 4. 16. Identify and sketch the quadric surface. a. b. c. d. e. f.
' + 3/
0 ./
+
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2 1/
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− =1 %4 4* = ' + 4) 9* − 4) − 9' = 36 * =) −' 4* = ' + 2) 0
+
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2