# Dot Product Vectors Worksheet

Dot Product Vectors Worksheet
• Page 1
1.
Find < 5$a$, 8$b$ > · < 5$a$ + 8$b$, 5$a$ - 8$b$ >.
 a. 25$a$2 - 64$b$2 + 80$a$$b$ b. 25$a$2 - 64$b$2 + 40$a$$b$ c. 25$a$2 - $b$2 + 80$a$$b$ d. 25$a$2 - $b$2 + 40$a$$b$

#### Solution:

< 5a, 8b > · < 5a + 8b, 5a - 8b > = 5a(5a + 8b) + 8b(5a - 8b)
[Use the definition of dot product.]

= 25a2 + 40ab + 40ba - 64b2

= 25a2 - 64b2 + 80ab
[As ab = ba.]

2.
Let u = < 6, 9 > and v = < 3, 5 >. Which of the following is equal to projvu?
 a. <$\frac{189}{34}$, $\frac{315}{34}$> b. <315 , 189 > c. <$\frac{3}{34}$, $\frac{5}{34}$> d. <189 , 315 >

#### Solution:

u = < 6, 9 > and v = < 3, 5 > are the two vectors.

|v| = 32+52 = 34
[Find the magnitude of v.]

u · v = < 6, 9 > · < 3, 5 >
[Find the dot product of u, v.]

= 6(3) + 9(5) = 63
[Use the definition of dot product.]

projvu = (uv|v |2) v
[Use the definition of Projvu.]

= (63 / 34) < 3, 5 >
[Substitute the values of u · v, |u| and |v|.]

= < 189 / 34, 315 / 34>
[Simplify.]

3.
Let u, v be two nonzero vectors. Which of the following is correct?
 a. projvu = (u · v) b. projvu = ()v c. Projection of u in the perpendicular direction of v = u - ()v d. Both B and C

#### Solution:

u, v are two nonzero vectors.

Projection of u on v = (uv|v |2)v
[Use the definition of Projvu.]

Projection of u in the perpendicular direction of v = u - (uv|v |2)v.
[Use the definition of projection of u in the perpendicular direction of v.]

4.
Let u = (cos 70°) $i$ + (sin 70°) $j$ and v = (cos 10°) $i$ + (sin 10°) $j$. Which of the following is the angle between the vectors u, v?
 a. 10° b. 60° c. 80° d. 70°

#### Solution:

u = (cos 70°) i + (sin 70°) j and v = (cos 10°) i + (sin 10°) j are the two vectors.

|u| = cos2 70° + sin2 70° = 1
[Use the formula to find the magnitude of u.]

|v| = cos2 10° + sin2 10° = 1
[Use the formula to find the magnitude of v.]

u · v = (cos 70°)(cos 10°) + (sin 70°)(sin 10°)
[Use the definition of the dot product to find u · v.]

= cos (70° - 10°)
[Use cos A cos B + sin A sin B = cos (A - B).]

= cos 60° = 12

Let θ be the angle between u and v.

So, cos θ = u  v|u||v|.
[Write the formula to find cos θ.]

= 12(1)(1) = 12
[Substitute the values of u · v, |u|, |v|.]

θ = Cos-1(12) = 60°
[Solve for θ.]

5.
Which of the following is correct for a vector u of a plane?
 a. u = (u · $i$) + (u · $j$) b. u = (u · $i$)$j$ + (u · $j$)$i$ c. u = (u · $j$) $i$ d. u = (u · $i$)$i$ + (u · $j$)$j$

#### Solution:

Let u = <x, y>

i = <1, 0> and j = <0, 1> are the standard unit vectors.

u · i = <x, y> · <1, 0> = x

u · j = <x, y> · <0, 1> = y

(u · i)i + (u · j)j = xi + yj = u

(u · i)j + (u · j)i = xj + yiu

u · i + u · j = x + yu

(u · j)i = yiu

So, (u · i)i + (u · j)j = u for any vector u of a plane.

6.
Let u = < sin 3 $\alpha$, cos 3 $\alpha$ >, v = < cos 6$\beta$, sin 6$\beta$ >. Which of the following is the value of u · v?
 a. sin (3α + 6β) b. cos (3α - 6β) c. sin (3α - 6β) d. cos (3α + 6β)

#### Solution:

u = < sin 3 α, cos 3 α >, v = < cos 6β, sin 6β > are the two vectors.

u · v = < sin 3 α, cos 3 α> · < cos 6β, sin 6β >

= (sin 3 α)(cos 6β) + (cos 3 α)(sin 6β)
[Use the definition of dot product.]

= sin (3 α + 6β)
[Use sin A cos B + cos A sin B = sin(A + B).

7.
Let u, v be two vectors. Which of the following is the value of |(3u + 6v)|2?
 a. |3u|2 + |6v|2 + 2(3u · 6v) b. |3u|2 - |6v|2 c. |3u|2 + |6v|2 + 2|3u| |6v| d. |3u|2 + |6v|2

#### Solution:

u, v are the two vectors.

|(3u + 6v)|2 = (3u + 6v) · (3u + 6v)
[For any vector x, |x|2 = x · x.]

= (3u + 6v) · 3u + (3u + 6v) · 6v
[For any three vectors x, y, z, x · (y + z) = x · y + x · z.]

= 3u · 3u + 6v · 3u + 3u · 6v + 6v · 6v

= |3u|2 + 3u · 6v + 3u · 6v + |6v|2
[For any vectors x, y, x · y = y · x.]

= |3u|2 + |6v|2 + 2(3u · 6v)

8.
Let u, v be two vectors. Which of the following is the value of (5u + 10v) · (5u - 10v)?
 a. |5u|2 - |10v|2 b. |5u + 10v|2 c. |5u - 10v|2 d. |5u|2 + |10v|2

#### Solution:

u, v are the two vectors.

(5u + 10v) · (5u - 10v) = 5u · (5u - 10v) + 10v · (5u - 10v)
[For any three vectors x, y, z, (x + y) · z = x · z + y · z.]

= 5u · 5u - 5u · 10v + 10v · 5u - 10v · 10v

= |5u|2 - 5u · 10v + 5u · 10v - |10v|2
[For any vectors x, y, x · y = y · x.]

= |5u|2 - |10v|2
[For a vector x, x - x = 0.]

9.
Let u, v be two vectors. Which of the following is correct?
 a. - |u| |v| ≤ u · v < |u| |v| b. - |u| |v| ≤ u · v ≤ |u| |v| c. - |u| |v| ≥ u · v ≥ |u| |v| d. - |u| |v| < u · v < |u| |v|

#### Solution:

u, v are two vectors. Let θ be the angle between u and v.

u · v = |u| |v| cos θ
[Use the definition of dot product.]

-1 ≤ cos θ ≤ 1
[Range of cos θ.]

- |u| |v| ≤ |u| |v| cos θ ≤ |u| |v|
[Multiply the inequality by |u| |v|, which is positive.]

- |u| |v| ≤ u · v ≤ |u| |v|
[Use u · v = |u| |v| cos θ.]

10.
Let u, v be any two nonzero vectors. Which of the following is correct?
 a. u · v is maximum if the angle between u, v is 180°. b. u · v is minimum if the angle between u, v is 0°. c. u · v is independent of the angle between u, v. d. u · v is maximum if the angle between u, v is 0°.

#### Solution:

u, v are two nonzero vectors. Let θ be the angle between them.

u · v = |u| |v| cos θ
[Use the definition of dot product.]

If θ = 0°, u · v = |u| |v| cos 0° = |u| |v|

If θ = 180°, u · v = |u| |v| cos 180° = - |u| |v|

So, u · v is maximum if the angle between them is 0°.