Question 1

Let $$\mathbf{u} = \langle 3, -1 \rangle$$ and $$\mathbf{v} = \langle -3, 4 \rangle. $$Find $$-\mathbf{u} + 2\mathbf{v}.$$

Accepted Answer

$$\langle -9, 9 \rangle$$

Question 2

If $$A is $$the point $$(-1, -7)$$ and $$B is $$the point $$(4, -14)$$, express the vector $$\overrightarrow{AB} in $$the form $$ai + bj.$$

Accepted Answer

$$\mathbf{v} = 5i - 7j$$

Question 3

Given $$\mathbf{u} = -2i - 7j$$ and $$\mathbf{v} = -4i - 21j$$, find $$|\mathbf{v} - \mathbf{u}|.$$

Accepted Answer

$$10\sqrt{2}$$

Question 4

Let $$\mathbf{u} = 6i + j$$, $$\mathbf{v} = i + j$$, and $$\mathbf{w} = i - j. $$Find scalars $$a$$ and $$b$$ such that $$\mathbf{u} = a\mathbf{v} + b\mathbf{w}.$$

Accepted Answer

$$a = 7/2$$, $$b = 5/2$$

Question 5

A force of magnitude $$8$$ pounds pulls on a suitcase at an angle of $$30^\circ$$ with the ground. Express the force in terms of its $$i$$ and $$j$$ components.

Accepted Answer

$$6.928i + 4.000j$$

Question 6

An audio speaker weighing $$50$$ pounds hangs from the ceiling by two cables at $$45^\circ$$ and $$35^\circ$$ angles. To two decimal places, what is the tension in the two cables?

Accepted Answer

Tension in right cable: $$35.90 lb$$; tension in left cable: $$41.59 lb$$

Question 7

Find the distance between the points $$P_1(1$$, -1, -2)$$ and P_2$(5, -6, -5).$$

Accepted Answer

$$5\sqrt{2}$$

Question 8

If $$\mathbf{v} = \langle -1, 6, 0 \rangle$$, find $$|\mathbf{v}|.$$

Accepted Answer

$$\sqrt{37}$$

Question 9

Find the position vector for the vector having initial point $$P = (-1, -3, 0)$$ and terminal point $$Q = (4, 3, -3).$$

Accepted Answer

$$\mathbf{v} = 5i + 6j - 3k$$

Question 10

Express $$2\mathbf{u} - 6\mathbf{v} in $$the form $$ai + bj + ck if$$ $$\mathbf{u} = (1, 1, 0)$$ and $$\mathbf{v} = (3, 0, 1).$$

Accepted Answer

$$-16i + 2j - 6k$$

Question 11

Find an equation for the sphere with center $$(-8, 10, 0)$$ and radius $$5.$$

Accepted Answer

$$x^2$$ + $$y^2$$ + $$z^2 - 16x + 20y = 139$

Question 12

Give a geometric description of the set of points whose coordinates satisfy $$x^2$$ + $$y^2$$ + $$z^2$$ \u003e 1$$.

Accepted Answer

All points outside the sphere of radius $$1$$

Question 13

Find the center and radius of the sphere $$x^2$$ + $$y^2$$ + $$z^2 - 18x - 10y - 6z = -15$.

Accepted Answer

Center: $$(9, 5, 3)$$, Radius: $$10$$

Question 14

A drone flies from point $$A(2, 3, 1) to $$point $$B(5, 7, 4). $$What is the displacement vector?

Accepted Answer

$$\langle 3, 4, 3 \rangle$$

Question 15

A boat travels $$10 km $$east and then $$6 km $$north. What is the magnitude of its displacement?

Accepted Answer

$$\sqrt{136} km$$

Question 16

If $$\mathbf{a} = \langle 2, -3, 4 \rangle$$ and $$\mathbf{b} = \langle -1, 5, 2 \rangle$$, find $$\mathbf{a} + \mathbf{b}.$$

Accepted Answer

$$\langle 1, 2, 6 \rangle$$

Question 17

A force $$\mathbf{F} of$$ $$12 N $$acts at an angle of $$60^\circ$$ above the horizontal. What are its $$i$$ and $$j$$ components?

Accepted Answer

$$6i + 10.39j$$

Question 18

Find the unit vector in the direction of $$\mathbf{v} = \langle 3, 4 \rangle.$$

Accepted Answer

$$\langle 0.6, 0.8 \rangle$$

Question 19

A car moves from $$(0, 0) to$$ $$(8, 6) in $$the plane. What is the direction angle $$	heta of $$its displacement vector with respect to the $$x-$$axis?

Accepted Answer

$$	heta = \arctan(6/8)$$

Vectors and Geometry in Calculus: Study Notes

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