Introduction to Vectors: Videos & Practice Problems

Understanding vector math is crucial in physics, particularly when combining vectors with direction. When adding parallel vectors, such as 3 meters right and 4 meters right, simply sum them to get 7 meters. However, for perpendicular vectors, like 3 meters right and 4 meters up, use the Pythagorean theorem: a2+b2=c2, leading to a resultant displacement of 5 meters. Mastering these concepts is essential for solving complex physics problems.

Introduction to Vector Math

In physics, understanding vector math is crucial as it involves both magnitude and direction, unlike scalars, which are simply numbers. When combining vectors, there are two primary scenarios to consider: parallel and perpendicular vectors.

For parallel vectors, such as moving 3 meters to the right and then 4 meters to the right, the total displacement is straightforward. You simply add the magnitudes: 3 + 4 = 7 meters. This principle holds true regardless of the direction, as long as the vectors are parallel.

However, when dealing with perpendicular vectors, the situation becomes more complex. For instance, if you walk 3 meters to the right and then 4 meters up, the total displacement is not simply the sum of these distances. Instead, you can visualize this scenario as forming a right triangle, where the two legs represent the movements. To find the hypotenuse, which represents the total displacement, you apply the Pythagorean theorem:

a^2 + b^2 = c^2

Here, a is 3 meters and b is 4 meters. Thus, the calculation becomes:

c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 meters.

This means that even though you moved 3 meters and 4 meters, the direct displacement is 5 meters in a diagonal direction.

In another example, if you walk 10 meters to the right and then 6 meters to the left, the total displacement is calculated as:

10 - 6 = 4 meters.

For a case involving perpendicular vectors again, if you walk 6 meters to the right and then 8 meters down, you would again use the Pythagorean theorem:

c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 meters.

In summary, vector math often involves visualizing movements as triangles and applying the Pythagorean theorem to find resultant displacements, especially when vectors are perpendicular. Mastering these concepts is essential for effectively working with vectors in physics.

Two perpendicular forces act on a box, one pushing to the right and one pushing up. An instrument tells you the magnitude of the total force is 13N. You measure the force pushing to the right is 12N. Calculate the force pushing up.

• A. 1 newton
• B. 5 newtons
• C. 12 newtons
• D. 18 newtons

Answer: (B)

Here’s what students ask on this topic:

What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude, such as mass or temperature. It is represented by a single number. A vector, on the other hand, has both magnitude and direction. Examples include displacement, velocity, and force. Vectors are often represented as arrows, where the length of the arrow indicates the magnitude and the direction of the arrow indicates the direction of the vector.

How do you add parallel vectors?

Adding parallel vectors is straightforward. If the vectors are in the same direction, you simply add their magnitudes. For example, if you have two vectors, 3 meters to the right and 4 meters to the right, the resultant vector is 7 meters to the right. If the vectors are in opposite directions, you subtract the smaller magnitude from the larger one. For instance, 10 meters to the right and 6 meters to the left result in a vector of 4 meters to the right.

How do you add perpendicular vectors?

To add perpendicular vectors, you use the Pythagorean theorem. For example, if you walk 3 meters to the right and then 4 meters up, you form a right triangle. The resultant vector is the hypotenuse of this triangle. Using the Pythagorean theorem: $a^2+b^2=c^2$, where $a$ is 3 meters and $b$ is 4 meters. The resultant vector $c$ is $\sqrt{3^2+4^2}=\; 5$ meters.

What is the Pythagorean theorem and how is it used in vector math?

The Pythagorean theorem is a fundamental principle in geometry that states: $a^2+b^2=c^2$, where $a$ and $b$ are the lengths of the legs of a right triangle, and $c$ is the length of the hypotenuse. In vector math, it is used to find the resultant vector when adding perpendicular vectors. For example, if you have vectors of 3 meters and 4 meters at right angles, the resultant vector is $\sqrt{3^2+4^2}=\; 5$ meters.

How do you calculate the resultant displacement of two vectors?

To calculate the resultant displacement of two vectors, you need to consider their directions. For parallel vectors, simply add or subtract their magnitudes. For perpendicular vectors, use the Pythagorean theorem. For example, if you walk 6 meters to the right and then 8 meters down, you form a right triangle. The resultant displacement is the hypotenuse, calculated as: $\sqrt{6^2+8^2}=\; 10$ meters.