Adding Vectors Graphically: Videos & Practice Problems

Vectors can be added graphically using the tip-to-tail method, where the resultant vector represents the shortest path from the start to the end point. The magnitude of the resultant can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$. When subtracting vectors, one vector is reversed, and the same principles apply. Multiplying vectors by scalars changes their magnitude but not their direction, affecting calculations of resultant vectors in physics.

Adding Vectors Graphically

Vectors are fundamental in physics, representing quantities that have both magnitude and direction, often visualized as arrows. To add vectors graphically, the tip-to-tail method is commonly used, where the tail of one vector is placed at the tip of another. This method allows for a clear visualization of the resultant vector, which is the shortest path from the starting point of the first vector to the endpoint of the last vector.

For example, if you walk 3 meters to the right and then 4 meters up, you can visualize this as two vectors: one pointing right (3 meters) and another pointing up (4 meters). By connecting these vectors tip to tail, you form a right triangle where the resultant vector can be calculated using the Pythagorean theorem. The magnitude of the resultant vector \( c \) is given by:

\[ c = \sqrt{a^2 + b^2} \]

In this case, \( c = \sqrt{3^2 + 4^2} = 5 \) meters. This resultant vector represents the direct displacement from the starting point to the endpoint.

When adding vectors that are not perpendicular, the same tip-to-tail method applies. For instance, if vector A is 2 meters to the right and 1 meter up, and vector B is 1 meter to the right and 3 meters up, you can connect them tip to tail to find the resultant vector. Regardless of the order in which you add the vectors (A + B or B + A), the resultant vector remains the same, demonstrating the commutative property of vector addition.

To further illustrate, if you add vector B first (1 meter right, 3 meters up) and then vector A (2 meters right, 1 meter up), you will still arrive at the same endpoint, confirming that the order of addition does not affect the resultant vector.

In another example, if vector A is combined with vector B, where B is 1 meter to the left and 4 meters down, you can again use the tip-to-tail method to find the resultant vector. By visualizing the vectors and applying the Pythagorean theorem, you can determine the magnitude of the resultant vector. If the legs of the triangle formed are both 3 meters, the magnitude of the resultant vector \( c \) is calculated as:

\[ c = \sqrt{3^2 + 3^2} = \sqrt{18} \approx 4.24 \text{ meters} \]

Understanding how to add vectors graphically and the properties of vector addition is crucial for solving problems in physics, as it provides a clear method for visualizing and calculating resultant displacements.

A delivery truck travels 8 miles in the +x-direction, 5 miles in the +y-direction, and 4 miles again in the +x-direction. What is the magnitude (in miles) of its final displacement from the origin?

• A. 17 miles
• B. 13 miles
• C. 9,0 miles
• D. 6.4 miles

Answer: (B)

Adding 3 Vectors

To find the magnitude of a resultant vector formed by the addition of three vectors originating from the same point, we can utilize the tip-to-tail method. This involves connecting the vectors sequentially, starting from the origin. The resultant vector is then determined by drawing a straight line from the start of the first vector to the end of the last vector.

In this example, we have three vectors: vector A, vector B, and vector C. Vector B moves 3 units to the left and 5 units up, while vector C moves 5 units to the left and 2 units down. By placing vector B tip to tail from the end of vector A, we can visualize its new position. Similarly, vector C is added tip to tail from the end of vector B.

After positioning all vectors correctly, the resultant vector D can be drawn directly from the origin to the endpoint of vector C. To calculate the magnitude of this resultant vector, we can form a right triangle where the legs correspond to the horizontal and vertical distances traveled. In this case, both legs measure 6 units.

To find the magnitude, we apply the Pythagorean theorem, which states:

\( c = \sqrt{a^2 + b^2} \)

Here, \( a \) and \( b \) are the lengths of the legs of the triangle. Substituting the values:

\( c = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 \)

Thus, the magnitude of the resultant vector D is approximately 8.5 units. If the vectors represent physical quantities such as meters, the final answer should include the appropriate units, indicating that the magnitude of the resultant vector is 8.5 meters.

Subtracting Vectors Graphically

Understanding vector subtraction is essential in physics, as it builds on the principles of vector addition. When subtracting vectors, the process is similar to addition, but one of the vectors is reversed. This means that vector subtraction can be conceptualized as adding a vector to the negative of another vector.

To illustrate, consider two vectors, A and B. When adding these vectors, they are connected tip to tail, and the resultant vector C is the straight line from the start of A to the end of B. For example, if A is represented as moving 2 units to the right and 1 unit up, and B moves 1 unit to the right and 3 units up, the resultant vector can be calculated using the Pythagorean theorem:

$$C = \sqrt{(2^2 + 3^2)} = \sqrt{13} \approx 3.61$$

However, when performing vector subtraction, such as A - B, we can think of it as A + (-B). The negative of vector B simply reverses its direction. If B moves 1 unit to the right and 3 units up, then -B will move 1 unit to the left and 3 units down. The resultant vector is then calculated by adding A and -B tip to tail.

For instance, if we have A as 2 units right and 1 unit up, and -B as 1 unit left and 3 units down, the resultant vector can be visualized and calculated as follows:

1. Start at the origin and draw A.
2. From the end of A, draw -B.
3. The resultant vector C is the straight line from the start to the end of -B.

Using the Pythagorean theorem again, if the horizontal displacement is 1 unit (2 right - 1 left) and the vertical displacement is 2 units (1 up - 3 down), the magnitude of the resultant vector is:

$$C = \sqrt{(1^2 + 2^2)} = \sqrt{5} \approx 2.24$$

It is crucial to note that the order of subtraction matters. For example, B - A would yield a different resultant vector than A - B, even though the magnitudes may be the same. This is because the direction of the resultant vector will differ based on the order of the vectors involved.

In summary, vector subtraction involves reversing the direction of one vector and then adding it to the other. This method allows for the calculation of resultant vectors in various scenarios, reinforcing the importance of understanding both vector addition and subtraction in physics.

Find the magnitude of the Resultant Vector D=C-B-A.

• A. 2.24
• B. 3.0
• C. 12.5
• D. 17

Answer: (A)

Adding Multiples of Vectors

Understanding vector addition and scalar multiplication is essential in physics and mathematics. When adding vectors, such as vectors a and b, the process involves aligning them tip to tail, with the resultant vector representing the shortest path from the start of the first vector to the end of the last. This method can be extended to include scalar multiples of vectors, such as 2a + 0.5b. Here, the scalar multipliers affect the magnitude of the vectors but not their direction.

For instance, if vector a has coordinates (3, 1), then 2a can be visualized as adding vector a to itself, resulting in the coordinates (6, 2). Conversely, 0.5b would halve the components of vector b. If b is (2, 4), then 0.5b becomes (1, 2). The resultant vector is then calculated by summing these adjusted vectors, leading to a new vector that can be analyzed for its magnitude.

The magnitude of a vector can be determined using the Pythagorean theorem. For a vector with components (x, y), the magnitude is given by the formula:

\( |C| = \sqrt{x^2 + y^2} \)

In the example of 2a + 0.5b, if the resultant vector has components (7, 4), the magnitude would be calculated as:

\( |C| = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.06 \)

When dealing with vector subtraction, the same principles apply. For example, in the expression 3a - 2b, 3a can be visualized as stacking three instances of vector a tip to tail, while -2b involves reversing the direction of vector b and stacking it twice. The resultant vector is then determined by connecting the start of the first vector to the end of the last vector, and its magnitude can be calculated similarly.

In summary, scalar multiplication alters the magnitude of a vector while maintaining its direction, with scalars greater than 1 increasing the length and those less than 1 decreasing it. The process of vector addition and subtraction remains consistent, allowing for the calculation of resultant vectors and their magnitudes using fundamental geometric principles.

Here’s what students ask on this topic:

How do you add vectors graphically using the tip-to-tail method?

To add vectors graphically using the tip-to-tail method, follow these steps: First, draw the first vector as an arrow. Then, place the tail of the second vector at the tip of the first vector. Continue this process for any additional vectors. The resultant vector is the arrow drawn from the tail of the first vector to the tip of the last vector. This resultant vector represents the shortest path from the start to the end point. The magnitude of the resultant can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$.

What is the resultant vector in vector addition?

The resultant vector in vector addition is the vector that represents the shortest path from the start of the first vector to the end of the last vector when vectors are added using the tip-to-tail method. It effectively combines the magnitudes and directions of all the vectors being added. The magnitude of the resultant vector can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$.

How do you subtract vectors graphically?

To subtract vectors graphically, you use the tip-to-tail method but reverse the direction of the vector being subtracted. For example, if you want to subtract vector B from vector A (A - B), you first draw vector A. Then, instead of adding vector B, you add the negative of vector B, which points in the opposite direction. The resultant vector is the shortest path from the start of vector A to the end of the negative vector B. The magnitude can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$.

What happens when you multiply a vector by a scalar?

When you multiply a vector by a scalar, the magnitude of the vector changes, but its direction remains the same. If the scalar is greater than 1, the vector's magnitude increases, making it longer. If the scalar is between 0 and 1, the vector's magnitude decreases, making it shorter. If the scalar is negative, the vector's direction reverses. For example, multiplying vector A by 2 (2A) doubles its length, while multiplying it by 0.5 (0.5A) halves its length.

Does the order of adding vectors affect the resultant vector?

No, the order of adding vectors does not affect the resultant vector. This property is known as the commutative property of vector addition. Whether you add vector A to vector B (A + B) or vector B to vector A (B + A), the resultant vector will be the same. The magnitude and direction of the resultant vector remain unchanged regardless of the order in which the vectors are added.