Chapter 7

For Questions 1 through 8, fill in the blank with an appropriate word. We use the law of cosines to find the missing parts of triangles for which we are given two _______ and the ______ between them, or for which we are given all three _______. In either case, a _________ triangle is determined.

For Questions 1 through 8, fill in the blank with an appropriate word. The dot product of two vectors is a ________ quantity. For this reason, it is sometimes called the _______ product.

If a vector V is in standard position and the tip of the vector corresponds to the point (a, b), then we can write the vector in component form as _______. The x-coordinate, a, is called the ___________ _________ of V, and the y-coordinate, b, is called the ________ ________ of V.

For Questions 1 through 8, fill in the blank with an appropriate word. The dot product of two vectors is also equal to the product of their __________ multiplied by the ______ of the angle between them.

To solve an oblique triangle given the case ASA, the first step is to find the missing so that the law of sines can be used.

A scalar is a _______ number. To multiply a vector in component form by a scalar, simply multiply each ________ by the scalar. This is called ________ multiplication.

For Questions 1 through 8, fill in the blank with an appropriate word. Perpendicular vectors are also said to be __________.

Multiplying a vector by a positive scalar will change the _______ of the vector but not its __________.

\text { State the formula for the semiperimeter of triangle } A B C: s=

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=150^{\circ}, b=30 \mathrm{ft}, a=10 \mathrm{ft}\); no solution

For Questions 1 through 8, fill in the blank with an appropriate word. Two nonzero vectors are perpendicular if and only if their _____ _______ is equal to ______.

The opposite of a vector is a vector with the ______ magnitude and __________ direction. To obtain the opposite of a vector, multiply the vector by ____.

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=30^{\circ}, b=40 \mathrm{ft}, a=15 \mathrm{ft}\); no solution

If \(B=120^{\circ}, C=20^{\circ}\), and \(c=28\) inches, find \(b\).

For Questions 1 through 8, fill in the blank with an appropriate word. The component of a force F that is oriented in the same direction as another vector d is called the _________ of F ____ d.

Find the semi perimeter of triangle \(A B C\). \(a=3 \mathrm{ft}, b=4 \mathrm{ft}, c=5 \mathrm{ft}\)

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=120^{\circ}, b=20 \mathrm{~cm}, a=30 \mathrm{~cm}\); one solution

For Questions 1 through 8, fill in the blank with an appropriate word. If a constant force F is applied to an object, and the resulting movement of the object is represented by the displacement vector d, then the work performed by the force is given by the ____ ______ of F and d.

The unit vector i points in the direction of the positive _____ and is called the _____ ________ vector. The unit vector j points in the direction of the positive _____ and is called the _____ ________ vector.

Find the semi perimeter of triangle \(A B C\). \(a=153 \mathrm{~cm}, b=174 \mathrm{~cm}, c=232 \mathrm{~cm}\)

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=30^{\circ}, b=10 \mathrm{~cm}, a=5 \mathrm{~cm}\); one solution

Find each of the following dot products $(3,4) \cdot\langle 5,5\rangle

Find the semi perimeter of triangle \(A B C\). \(a=2.1 \mathrm{~m}, b=2.3 \mathrm{~m}, c=3.9 \mathrm{~m}\)

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=60^{\circ}, b=18 \mathrm{~m}, a=16 \mathrm{~m}\); two solutions

If \(A=10^{\circ}, C=150^{\circ}\), and \(a=24 \mathrm{yd}\), find \(c\).

Find each of the following dot products $\langle 6,6\rangle \cdot\langle-3,5\rangle

$$ \text { Complete the formula for the law of cosines: } \cos A= $$ __________

The magnitude of \(\mathbf{V}=\langle a, b\rangle=a \mathbf{i}+b \mathbf{j}\) is given by _____.

Find the semi perimeter of triangle \(A B C\). \(a=33 \mathrm{yd}, b=45 \mathrm{yd}, c=6 \mathrm{yd}\)

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=20^{\circ}, b=45 \mathrm{~m}, a=25 \mathrm{~m}\); two solutions

If \(A=50^{\circ}, B=60^{\circ}\), and \(a=36 \mathrm{~km}\), find \(C\) and then find \(c\).

Find each of the following dot products \((-23,4) \cdot(15,-6)\)

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=120 \text { inches, } b=66 \text { inches, and } C=60^{\circ} \text {, find } c \text {. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(4,1)$$

Find all solutions to each of the following triangles: \(A=38^{\circ}, a=41 \mathrm{ft}, b=54 \mathrm{ft}\)

If \(B=40^{\circ}, C=70^{\circ}\), and \(c=82 \mathrm{~km}\), find \(A\) and then find \(a\).

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=48 \text { inches, } b=84 \text { inches, and } C=120^{\circ} \text {, find } c \text {. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(1,4)$$

Find all solutions to each of the following triangles: \(A=43^{\circ}, a=31 \mathrm{ft}, b=37 \mathrm{ft}\)

If \(A=52^{\circ}, B=48^{\circ}\), and \(c=14 \mathrm{~cm}\), find \(C\) and then find \(a\).

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=\mathrm{i}+\mathbf{j}, \mathrm{V}=\mathrm{i}-\mathrm{j}\)

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=13 \mathrm{yd}, b=14 \mathrm{yd} \text {, and } c=15 \mathrm{yd} \text {, find the largest angle. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(-5,2)$$

Find all solutions to each of the following triangles: \(A=112.2^{\circ}, a=43.8 \mathrm{~cm}, b=22.3 \mathrm{~cm}\)

If \(A=33^{\circ}, C=82^{\circ}\), and \(b=44 \mathrm{~cm}\), find \(B\) and then find \(c\).

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). $\mathbf{U}=-\mathbf{i}+\mathbf{j}, \mathrm{V}=-\mathrm{i}-\mathbf{j

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=22 \mathrm{yd}, b=24 \mathrm{yd} \text {, and } c=29 \mathrm{yd} \text {, find the largest angle. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(-2,5)$$

Find all solutions to each of the following triangles: \(A=132.4^{\circ}, a=27.3 \mathrm{~cm}, b=50.2 \mathrm{~cm}\)

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=-3 \mathbf{i}, \mathbf{V}=5 \mathbf{j}\)