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S. 18.02 Solutions to Exercises
1. Vectors and Matrices
1A. Vectors
√ √
1A-1 a) |A| = 3; dir A = A= 3 b) |A| = 3; dir A = A=3
c) |A| = 7; dir A = A=7
1A-2 1=25+1=25+c2 = 1 ⇒ c = ±√23=5
1A-3 a) A = −i −2j +2k ; |A| = 3; dir A = A=3:
b) A = |A| dir A = 2i + 4j − 4k . Let P be its tail and Q its head. Then
OQ = OP + A = 4j −3k ; therefore Q = (0;4;−3).
1A-4 a) OX = OP + PX = OP + 1(PQ) = OP + 1(OQ−OP) = 1(OP + OQ)
2 2 2
b) OX = sOP + rOQ; replace 1 by r in above; use 1 − r = s.
2
√
1A-5 A = 3 3i + 3 j . The condition is not redundant since there are two vectors of
2 2 o
length 3 making an angle of 30 with i .
√ √ √
1A-6 wind w= 50(−i − j )= 2); v + w = 200j ⇒ v = 50= 2i +(200+50= 2)j .
2 2 ′
1A-7 a) bi −aj b) −bi + aj c) (3=5) +(4=5) = 1; j = −(4=5)i +(3=5)j
1A-8 a) is elementary trigonometry;
√ 2 2 2
b) cosα = a= a + b + c , etc.; dir A = (−1=3;2=3;2=3)
c) if t; u; v are direction cosines of some A, then ti +uj +v k = dir A, a unit vector,
so t2 +u2 +v 2 = 1; conversely, if this relation holds, then ti +uj +v k = u is a unit vector,
so dir u = u and t;u;v are the direction cosines of u.
1A-9 Letting A and B be the two sides, the third side is B −A; the line B
1 1 1 B-A
joining the two midpoints is B − A, which = (B −A), a vector parallel
2 2 2 A
to the third side and half its length.
C
1A-10 Letting A;B;C,D be the four sides; then if the vectors are suitably D
A
oriented, we have A + B = C+D. B
The vector from the midpoint of A to the midpoint of C is 1C − 1A; similarly, the
2 2
vector joining the midpoints of the other two sides is 1B − 1D, and
2 2
A + B = C + D ⇒ C −A = B −D ⇒ 1(C −A)= 1(B −D) ;
2 2
thus two opposite sides are equal and parallel, which shows the figure is a parallelogram.
1A-11 Letting the four vertices be O;P;Q;R; with X on PR and Y on OQ,
OX = OP + PX = OP + 1PR R Q
2
= OP + 1(OR−OP) Y
2 X
= 1(OR+ OP) = 1OQ = OY;
therefore X = Y . 2 2 O P
1
2 S. 18.02 SOLUTIONS TO EXERCISES
1B. Dot Product
A· B 4+2 1 π 3 1 π
1B-1 a) cosθ = |A||B| = √ = √ ; θ = 4 b) cosθ = √ √ = 2; θ = 3 .
2·6 2 6· 6
1B-2 A · B = c − 4; therefore (a) orthogonal if c = 4,
√ c −4
b) cosθ = 2 √ ; the angle θ is acute if cosθ > 0, i.e., if c > 4.
c +5 6
1B-3 Place the cube in the first octant so the origin is at one corner P, and i ; j ; k are
three edges. The longest diagonal PQ = i + j + k ; a face diagonal PR = i + j .
PQ· PR 2 −1 �
a) cosθ = |PQ| · |PR| = √ √ ; θ = cos 2=3
3 2
PQ· i 1 −1 √
b) cosθ = |PQ||i | = √3 ; θ = cos 1= 3.
1B-4 QP = (a;0;−2); QR=(a;−2;2), therefore
a) QP· QR= a2 −4; therefore PQR is a right angle if a2 −4 = 0, i.e., if a = ±2.
a2 −4
b) cosθ = √ √ ; the angle is acute if cosθ > 0, i.e., if a2 − 4 > 0, or
a2 +4 a2 +8
|a| > 2, i.e., a > 2 or a < −2.
√
1B-5 a) F·u = −1= 3 b) u =dir A = A=7, so F · u = −4=7
1B-6 After dividing by |OP|, the equation says cosθ = c, where θ is the angle between OP
and u; call its solution θ0 = cos−1 c: Then the locus is the nappe of a right circular cone
with axis in the direction u and vertex angle 2θ0. In particular this cone is
a) a plane if θ0 = π=2, i.e., if c = 0 b) a ray if θ0 = 0; π, i.e., if c = ±1.
c) Locus is the origin, if c > 1 or c < −1 (division by |OP| is illegal, notice).
√2
′ ′ √
1B-7 a) |i | = |j | = 2 = 1; a picture shows the system is right-handed.
′ √ ′ √
b) A· i = −1= 2; A· j = −5= 2; ′ ′
−i −5j
since they are perpendicular unit vectors, A = √2 .
′ ′ ′ ′
i − j i + j
c) Solving, i = √ ; j = √ ;
2 2
′ ′ ′ ′ ′ ′
2(i − j ) 3(i + j ) −i −5j
thus A = 2i −3j = √2 − √2 = √2 , as before.
′ ′ ′ ′ ′ ′
1B-8 a) Check that each has length 1, and the three dot products i · j ; i · k ; j · k
are 0; make a sketch to check right-handedness.
′ √ ′ ′ √ √ ′ √ ′
b) A· i = 3; A· j = 0; A· k = 6, therefore, A = 3i + 6k .
1B-9 Let u = dir A, then the vector u-component of B is (B· u)u. Subtracting it off gives
a vector perpendicular to u (and therefore also to A); thus
� �
B =(B·u)u + B −(B· u)u
1. VECTORS AND MATRICES 3
or in terms of A, remembering that |A|2 = A · A,
B· A � B· A �
B = + B −
A· A A A· A A
.
1B-10 Let two adjacent edges of the parallelogram be the vectors A and B; then the two
2
diagonals are A +B and A −B. Remembering that for any vector C we have C · C = |C| ,
the two diagonals have equal lengths
⇔ (A + B)· (A + B) = (A −B)· (A −B)
⇔ (A · A) + 2(A · B) + (B · B) = (A · A) − 2(A · B) + (B · B)
⇔ A· B = 0;
which says the two sides are perpendicular, i.e., the parallelogram is a rectangle.
1B-11 Using the notation of the previous exercise, we have successively,
(A + B)· (A −B) = A·A −B· B; therefore,
(A + B)· (A −B) = 0 ⇔ A· A = B· B;
i.e., the diagonals are perpendicular if and only if two adjacent edges have equal length, in
other words, if the parallelogram is a rhombus.
1B-12 Let O be the center of the semicircle, Q and R the two ends of the diameter, and
P the vertex of the inscribed angle; set A = QO = OR and B = OP; then |A| = |B|.
The angle sides are QP = A + B and PR = A − B; they are perpendicular since
(A + B)· (A −B) = A·A −B· B
= 0; since |A| = |B|:
1B-13 The unit vectors are ui = cosθi i + sinθi j , for i = 1;2; the angle between them is
θ2 −θ1. We then have by the geometric definition of the dot product
cos(θ2 −θ1) = u1 · u2 ;
|u1||u2|
= cosθ1 cosθ2 +sinθ1 sinθ2;
according to the formula for evaluating the dot product in terms of components.
1B-14 Let the coterminal vectors A and B represent two sides of the triangle, and let θ
be the included angle. Suitably directed, the third side is then C = A − B, and
|C|2 = (A −B)· (A −B) = A · A + B · B −2A · B
= |A|2 + |B|2 − 2|A||B|cosθ;
by the geometric interpretation of the dot product.
4 S. 18.02 SOLUTIONS TO EXERCISES
1C. Determinants
� � � �
� 1 4 � � 3 −4 �
1C-1 a) � � = −1−8 = −9 b) � � = −10.
� 2 −1� � −1 −2�
� �
� −1 0 4�
� �
1C-2 � 1 2 2 � = 2+0−8−(24+4+0)= −34:
� �
� 3 −2 −1 � � � � � � �
� 2 2� � 1 2� � 1 2�
a) By the cofactors of row one: = −1 � � − 0 · � � + 4 · � � = −34
� −2 −1 � � 3 −1 � � 3 −2 �
� � � � � �
� 2 2� � 0 4� � 0 4�
b) By the cofactors of column one: = −1· � � −1· � � +3· � � = −34.
� −2 −1 � � −2 −1 � � 2 2�
� �
� 1 2�
1C-3 a) � � = −3; so area of the parallelogram is 3, area of the triangle is 3/2
� 1 −1� � �
� 0 −3 �
b) sides are PQ = (0;−3); PR = (1;1); � � = 3; so area of the parallelogram
is 3, area of the triangle is 3/2 � 1 1�
� �
� 1 1 1 �
� � 2 2 2 2 2 2
1C-4 �x1 x2 x3 � = x2x3 + x1x2 + x1x3 −x1x2 −x 2x3 −x1x3
� 2 2 2 �
� x1 x2 x3 � 2 2 2 2 2 2
(x −x )(x −x )(x −x ) = x x −x x −x x x +x x −x x +x x x +x x −x x :
1 2 1 3 2 3 1 2 1 3 1 3 2 1 3 2 1 2 1 3 2 3 2 3
Two terms cancel, and the other six are the same as those above, except they have the
opposite sign.
� � � �
� x1 y1 � � x1 y1 �
1C-5 a) � � = x1y2 + ax1y1 −x2y1 −ay1x1 = � � :
� x2 + ax1 y2 + ay1 � � x2 y2�
b) is similar.
1C-6 Use the Laplace expansion by the cofactors of the first row.
1C-7 The heads of two vectors are on the unit circle. The area of the parallelogram they
span is biggest when the vectors are perpendicular, since area = absinθ = 1 · 1 · sinθ; and
sinθ has its maximum when θ = π=2.
� �
� x1 y1 �
Therefore the maximum value of � � = area of unit square = 1.
� x2 y2 �
1C-9 PQ = (0;−1;2); PR = (0;1;−1); PS = (1;2;1);
� �
� 0 −1 2�
� �
volume parallelepiped = ±� 0 1 −1 � = ±(−1) = 1 :
� �
� 1 2 1�
vol. tetrahedron = 1(base)(ht.) = 1 · 1 (p’piped base)(ht.) = 1(vol. p’piped) = 1=6.
3 3 2 6
1C-10 Thinking of them all as origin vectors, A lies in the plane of B and C, therefore
the volume of the parallelepiped spanned by the three vectors is zero.
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