An engineer finds the angle between two force vectors using the dot product. Given Ax (Ax) = 7.5000; Ay (Ay) = 4.5000; Bx (Bx) = 4.5000; By (By) = 9.0000, determine the angle between vectors (theta) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Vectors — dot product and angle.
• Everything except theta is given, so isolate theta symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Vector dot products give the angle between two vectors through the relation A·B = |A||B|cos(theta).

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos\theta = \dfrac{\vec{A}\cdot\vec{B}}{|A||B|}$$

2. Step 2 — Rearrange symbolically for theta:

   $$\theta = \cos^{-1}\!\left(\dfrac{\vec{A}\cdot\vec{B}}{|A||B|}\right)$$

3. Step 3

   $$List the givens: Ax (Ax) = 7.5000, Ay (Ay) = 4.5000, Bx (Bx) = 4.5000, By (By) = 9.0000$$

4. Step 4 — Substitute the given values:

   $$\theta = \cos^{-1}\!\left(\dfrac{\vec{A}\cdot\vec{B}}{|A||B|}\right)$$

5. Step 5 — Evaluate:

   $$\theta = 32.4712\ \text{deg}$$

6. Step 6 — Check: returning theta = 32.4712 deg to

   $$\cos\theta = \dfrac{\vec{A}\cdot\vec{B}}{|A||B|}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 64.9424 — kept a factor of two that cancels in the correct rearrangement.
• 16.2356 — dropped that same factor in the other direction.
• 35.7183 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors

a wind vector resolved onto a roof-plane normal Given magnitude of vector A (A) = 4.5000 kN; magnitude of vector B (B) = 9.5000 kN; cosine of the included angle (c) = 0.5000, determine the dot product (D) in kN^2.

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for D:

   $$D = A B \cos\theta$$

3. Step 3 — List the givens: magnitude of vector A (A) = 4.5000 kN, magnitude of vector B (B) = 9.5000 kN, cosine of the included angle (c) = 0.5000.

4. Step 4 — Substitute the given values:

   $$D = 4.5000 9.5000 \cos\theta$$

5. Step 5 — Evaluate:

   D = 21.3750\ \text{kN^2}

6. Step 6 — Check: returning D = 21.3750 kN^2 to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 42.7500 — kept a factor of two that cancels in the correct rearrangement.
• 10.6875 — dropped that same factor in the other direction.
• 23.5125 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)

a displacement vector projected on a survey baseline Given dot product (D) = 376.2 kN^2; magnitude of vector B (B) = 7.0000 kN; cosine of the included angle (c) = 0.6000, determine the magnitude of vector A (A) in kN.

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except A is given, so isolate A symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for A:

   $$A = \dfrac{D}{B \cos\theta}$$

3. Step 3 — List the givens: dot product (D) = 376.2 kN^2, magnitude of vector B (B) = 7.0000 kN, cosine of the included angle (c) = 0.6000.

4. Step 4 — Substitute the given values:

   $$A = \dfrac{376.2}{7.0000 \cos\theta}$$

5. Step 5 — Evaluate:

   $$A = 89.5714\ \text{kN}$$

6. Step 6 — Check: returning A = 89.5714 kN to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 179.1 — kept a factor of two that cancels in the correct rearrangement.
• 44.7857 — dropped that same factor in the other direction.
• 98.5286 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)

two cable forces meeting at a tower anchor Given dot product (D) = 181.6 kN^2; magnitude of vector A (A) = 6.0000 kN; magnitude of vector B (B) = 25.0000 kN, determine the cosine of the included angle (c).

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except c is given, so isolate c symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for c:

   $$c = \dfrac{D}{A B}$$

3. Step 3 — List the givens: dot product (D) = 181.6 kN^2, magnitude of vector A (A) = 6.0000 kN, magnitude of vector B (B) = 25.0000 kN.

4. Step 4 — Substitute the given values:

   $$c = \dfrac{181.6}{6.0000 25.0000}$$

5. Step 5 — Evaluate:

   $$c = 1.2107$$

6. Step 6 — Check: returning c = 1.2107 to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.4213 — kept a factor of two that cancels in the correct rearrangement.
• 0.6053 — dropped that same factor in the other direction.
• 1.3317 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)

An engineer finds the angle between two force vectors using the dot product. Given Ax (Ax) = 5.0000; Ay (Ay) = 1.0000; Bx (Bx) = 6.0000; By (By) = 10.0000, determine the angle between vectors (theta) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Vectors — dot product and angle.
• Everything except theta is given, so isolate theta symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Vector dot products give the angle between two vectors through the relation A·B = |A||B|cos(theta).

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos\theta = \dfrac{\vec{A}\cdot\vec{B}}{|A||B|}$$

2. Step 2 — Rearrange symbolically for theta:

   $$\theta = \cos^{-1}\!\left(\dfrac{\vec{A}\cdot\vec{B}}{|A||B|}\right)$$

3. Step 3

   $$List the givens: Ax (Ax) = 5.0000, Ay (Ay) = 1.0000, Bx (Bx) = 6.0000, By (By) = 10.0000$$

4. Step 4 — Substitute the given values:

   $$\theta = \cos^{-1}\!\left(\dfrac{\vec{A}\cdot\vec{B}}{|A||B|}\right)$$

5. Step 5 — Evaluate:

   $$\theta = 47.7263\ \text{deg}$$

6. Step 6 — Check: returning theta = 47.7263 deg to

   $$\cos\theta = \dfrac{\vec{A}\cdot\vec{B}}{|A||B|}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 95.4526 — kept a factor of two that cancels in the correct rearrangement.
• 23.8632 — dropped that same factor in the other direction.
• 52.4989 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors

a wind vector resolved onto a roof-plane normal Given magnitude of vector A (A) = 3.0000 kN; magnitude of vector B (B) = 12.0000 kN; cosine of the included angle (c) = 0.6600, determine the dot product (D) in kN^2.

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for D:

   $$D = A B \cos\theta$$

3. Step 3 — List the givens: magnitude of vector A (A) = 3.0000 kN, magnitude of vector B (B) = 12.0000 kN, cosine of the included angle (c) = 0.6600.

4. Step 4 — Substitute the given values:

   $$D = 3.0000 12.0000 \cos\theta$$

5. Step 5 — Evaluate:

   D = 23.7600\ \text{kN^2}

6. Step 6 — Check: returning D = 23.7600 kN^2 to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 47.5200 — kept a factor of two that cancels in the correct rearrangement.
• 11.8800 — dropped that same factor in the other direction.
• 26.1360 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)

a displacement vector projected on a survey baseline Given dot product (D) = 315.2 kN^2; magnitude of vector B (B) = 5.0000 kN; cosine of the included angle (c) = 0.8400, determine the magnitude of vector A (A) in kN.

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except A is given, so isolate A symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for A:

   $$A = \dfrac{D}{B \cos\theta}$$

3. Step 3 — List the givens: dot product (D) = 315.2 kN^2, magnitude of vector B (B) = 5.0000 kN, cosine of the included angle (c) = 0.8400.

4. Step 4 — Substitute the given values:

   $$A = \dfrac{315.2}{5.0000 \cos\theta}$$

5. Step 5 — Evaluate:

   $$A = 75.0476\ \text{kN}$$

6. Step 6 — Check: returning A = 75.0476 kN to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 150.1 — kept a factor of two that cancels in the correct rearrangement.
• 37.5238 — dropped that same factor in the other direction.
• 82.5524 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)

two cable forces meeting at a tower anchor Given dot product (D) = 245.7 kN^2; magnitude of vector A (A) = 10.0000 kN; magnitude of vector B (B) = 6.0000 kN, determine the cosine of the included angle (c).

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except c is given, so isolate c symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for c:

   $$c = \dfrac{D}{A B}$$

3. Step 3 — List the givens: dot product (D) = 245.7 kN^2, magnitude of vector A (A) = 10.0000 kN, magnitude of vector B (B) = 6.0000 kN.

4. Step 4 — Substitute the given values:

   $$c = \dfrac{245.7}{10.0000 6.0000}$$

5. Step 5 — Evaluate:

   $$c = 4.0950$$

6. Step 6 — Check: returning c = 4.0950 to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 8.1900 — kept a factor of two that cancels in the correct rearrangement.
• 2.0475 — dropped that same factor in the other direction.
• 4.5045 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)

An engineer finds the angle between two force vectors using the dot product. Given Ax (Ax) = 3.5000; Ay (Ay) = 10.0000; Bx (Bx) = 3.0000; By (By) = 9.5000, determine the angle between vectors (theta) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Vectors — dot product and angle.
• Everything except theta is given, so isolate theta symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Vector dot products give the angle between two vectors through the relation A·B = |A||B|cos(theta).

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos\theta = \dfrac{\vec{A}\cdot\vec{B}}{|A||B|}$$

2. Step 2 — Rearrange symbolically for theta:

   $$\theta = \cos^{-1}\!\left(\dfrac{\vec{A}\cdot\vec{B}}{|A||B|}\right)$$

3. Step 3

   $$List the givens: Ax (Ax) = 3.5000, Ay (Ay) = 10.0000, Bx (Bx) = 3.0000, By (By) = 9.5000$$

4. Step 4 — Substitute the given values:

   $$\theta = \cos^{-1}\!\left(\dfrac{\vec{A}\cdot\vec{B}}{|A||B|}\right)$$

5. Step 5 — Evaluate:

   $$\theta = 1.7645\ \text{deg}$$

6. Step 6 — Check: returning theta = 1.7645 deg to

   $$\cos\theta = \dfrac{\vec{A}\cdot\vec{B}}{|A||B|}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.5290 — kept a factor of two that cancels in the correct rearrangement.
• 0.8822 — dropped that same factor in the other direction.
• 1.9409 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors

a wind vector resolved onto a roof-plane normal Given magnitude of vector A (A) = 6.5000 kN; magnitude of vector B (B) = 22.0000 kN; cosine of the included angle (c) = 0.6800, determine the dot product (D) in kN^2.

Start with the thinking

• The governing relation printed in this handbook section is Dot product and the angle between vectors.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two force vectors act at a joint and the projection of one onto the other is required.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A \cdot B = |A||B|\cos\theta$$

2. Step 2 — Rearrange symbolically for D:

   $$D = A B \cos\theta$$

3. Step 3 — List the givens: magnitude of vector A (A) = 6.5000 kN, magnitude of vector B (B) = 22.0000 kN, cosine of the included angle (c) = 0.6800.

4. Step 4 — Substitute the given values:

   $$D = 6.5000 22.0000 \cos\theta$$

5. Step 5 — Evaluate:

   D = 97.2400\ \text{kN^2}

6. Step 6 — Check: returning D = 97.2400 kN^2 to

   $$A \cdot B = |A||B|\cos\theta$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 194.5 — kept a factor of two that cancels in the correct rearrangement.
• 48.6200 — dropped that same factor in the other direction.
• 107.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Vectors (Dot Product)