M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
Interphase Calculus III Worksheet
Instructor: Samuel S. Watson
01 July 2015
Topics: Integration by trigonometric substitution, dot product, cross product, triple scalar product.
1. Recall that for all real numbers θ , we have sin2 θ + cos2 θ = 1. Divide both sides of this equation
by cos2 θ to derive a relationship between tan θ and sec θ . Divide by sin2 θ to derive a relationship
between cot θ and csc θ .
2. Consider substituting x = a f (θ ) into the expression x2 + 16, where a is a constant and f is one
of the trig functions. For which values
of a and f does the expression x2 + 16 simplify to a single
Z
1
term? Use your observation to find
dx.
2
( x + 16)3/2
3. Integrate
Z
√
1
x2 + 2x
dx using trigonometric substitution.
4. We denote by Rn the set of order n-tuples of real numbers. For example,
R3 = {( x, y, z) : x, y, z ∈ R}.
We define addition of vectors componentwise, and we can multiply vectors by real numbers
(called scalars in this context) componentwise:
( x1 , y1 ) + ( x2 , y2 ) = ( x1 + x2 , y1 + y2 )
λ( x, y) = (λ x, λy).
We denote vectors with boldface notation of an arrow: u or ~u. Use the definitions above to verify
that
λ(u + v) = λu + λv, and
u + v = v + u,
5. We depict a vector u = ( x, y) graphically by drawing an arrow from the origin to the point
( x, y), or any translate of this arrow. Define u = (1, 2) and v = (3, 1) and confirm that u, v, and
u + v fit together to form a triangle.
6. The dot product of two vectors is defined by
( x1 , y1 , z1 ) · ( x2 , y2 , z2 ) = x1 x2 + y1 y2 + z1 z2 ,
q
and the norm (or length) of a vector is defined by |( x, y, z)| = x2 + y2 + z2 .
Verify that u · (v + w) = u · v + u · w and that u · u = |u|2 .
7. Use the diagram below and the law of cosines to show that u · v = |u||v| cos θ , where θ is the
angle between u and v.
u−v
v
θ
u
8. Show that the vectors (6, 3) and (2, −4) are perpendicular, while (8, 9) and (−10, 9) are not.
9. The cross product of u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) is defined to be
u × v = ( u2 v3 − u3 v2 , v1 u3 − u1 v3 , u1 v2 − u2 v1 )
Confirm that u × v is perpendicular to u.
10. Verify that |u × v|2 = |u|2 |v|2 − (uv)2 . Use this fact to show that
|u × v| = |u||v| sin θ,
where θ is the angle between u and v. Relate |u × v| to the area of the parallelogram spanned by
u and v.
11. Find the equation of the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1). (Hint:
find two vectors contained in the plane, and the cross them to find a vector perpendicular to the
plane.)
12. Find the volume of the parallelepiped spanned by (3, 4, 1), (−2, 4, 0), and (−5, 5, 2). (Hint:
first find the area of the base, then figure out how to use a dot product to multiply by the height.)
13. Challenge problem. (#34 on page 796 in your book) Show that the vector (a × b) × (c × d) can
be written in the form r1 a + r2 b where r1 and r2 are scalars.