3. Vectors

Adding Vectors by Components

Adding Vectors: How to Find the Resultant of Three or More Vectors

FIGURE P3.43 shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving? Give the direction as an angle below the negative x-axis. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/631d351c39ab4d14-1676668315030.jpg" alt="" />

A position vector in the first quadrant has an x-component of 6 m and a magnitude of 10 m. What is the value of its y-component?

Adding Vector Components Example | Physics with Professor Matt Anderson | M3-11

Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward’s displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

A runner is training for an upcoming marathon by running around a 100-m-diameter circular track at constant speed. Let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. The runner starts at (x,y) = (50m, 0m) and runs 2.5 times around the track in aclockwise direction. What is his displacement vector? Give your answer as a magnitude and direction.

While vacationing in the mountains you do some hiking. In the morning, your displacement is Sₘₒᵣₙᵢₙ₉ = (2000 m, east) ＋ (3000 m, north) ＋(200 m, vertical). Continuing on after lunch, your displacement is Sₐբₜₑᵣₙₒₒₙ = (1500 m, west) ＋ (2000 m, north) ─ (300 m, vertical).

b. What is the magnitude of your net displacement for the day?

Vector Addition By Components

Vectors - Finding the components of a vector

Let <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mn>2.0</mn><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>, <math display="inline"><mrow><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mo>&#x2212;</mo><mn>7.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover></mrow></math>, and <math display="inline"><mrow><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mn>3.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>. What is the direction of <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover></mrow></math>? Express your answer in degrees counter clockwise from the positive x axis.

Let <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mn>2.0</mn><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>, <math display="inline"><mrow><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mo>&#x2212;</mo><mn>7.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover></mrow></math>, and <math display="inline"><mrow><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mn>3.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>. What is the magnitude of <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover></mrow></math>?

Anderson Video - Adding Vector Components Example

A velocity vector 40 degrees below the positive x-axis has a y-component of -10 m/s. What is the value of its x-component?

Adding Vectors

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zig-zag paths at a constant speed of 20 μm/s. FIGURE P3.42 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium’s average velocity for the entire trip? <img src="https://lightcat-files.s3.amazonaws.com/problem_images/93e1dbb5a85bb398-1677783838193.jpg" alt="" />

A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components. <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/93083a9e0fc31bff-1678190867393.jpg" style="width: 300px;">

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (d) the vector difference B - A <img src="https://lightcat-files.s3.amazonaws.com/problem_images/c27460d9a4ad6484-1673442520803.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/1a1966ce78d4fae9-1678190776065.jpg" style="width: 182px;">

Anderson Video - Adding Vector Components

Anderson Video - Adding Vectors Example

Vector Component Addition and Subtraction

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (a) the vector sum A + B <img src="https://lightcat-files.s3.amazonaws.com/problem_images/91ff45272b60d5fe-1673475545736.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/1468aa562fe00b6b-1678190695545.jpg" style="width: 178px;">

While vacationing in the mountains you do some hiking. In the morning, your displacement is Sₘₒᵣₙᵢₙ₉ = (2000 m, east) ＋ (3000 m, north) ＋(200 m, vertical). Continuing on after lunch, your displacement is Sₐբₜₑᵣₙₒₒₙ = (1500 m, west) ＋ (2000 m, north) ─ (300 m, vertical).

a. At the end of the hike, how much higher or lower are you compared to your starting point?

Physics Vector Addition (Algebraic)

Adding Vector Components | Physics with Professor Matt Anderson | M3-07

FIGURE P3.26 shows vectors A and B. Find D = 2A ＋B Write your answer in component form.
<img src="https://lightcat-files.s3.amazonaws.com/problem_images/9844d2ece039903b-1676567203420.jpg" alt="" />

Anderson Video - Displacement Vectors and Adding Vectors

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (c) the vector difference A - B <img src="https://lightcat-files.s3.amazonaws.com/problem_images/78f96ca97e8dd6a9-1670246445278.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/ee074203d82f1a52-1678190795461.jpg" style="width: 171px;">

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