2.16:
Dot Product: Problem Solving
Consider a rod fixed to a wall, which can be pulled by a chain by applying a force at one of its ends. The position of the rod is defined using a three-dimensional coordinate system.
The angle theta between the force vector and the rod, and the projection of force along the rod needs to be determined.
First, the position vectors for the two ends of the rod are defined. Then the position vector along the rod is determined.
The next step determines the magnitude of the position vector rAB and the force vector.
Now, the dot product of the position vector with the force vector is determined by multiplying the components of the two vectors. Angle theta is then estimated as the inverse cosine function of the ratio of the dot product and the product of magnitudes of the two vectors.
The projection of the force along the rod can be determined as the product of the magnitude of force and the cosine of theta.
2.16:
Dot Product: Problem Solving
The dot product is a powerful tool in problem-solving involving vectors, given that the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them measured anti-clockwise. Solving problems involving the dot product requires understanding its properties and developing a step-by-step process to solve them. Here are the main steps to follow when solving any general problem involving the dot product:
Identify the problem: Start by reading the problem and identifying the question that needs to be answered. This will enable you to determine the purpose and direction for solving the problem.
Define the vectors: List the given vectors and represent them in the Cartesian or component form.
Decide which operation to use: The dot product is appropriate when the problem involves finding the angle between two vectors, calculating the component of a vector along a given direction, testing orthogonality, or finding the projection of one vector onto another vector. Ensure that the problem requires the use of the dot product before proceeding.
Calculate the dot product: Multiply the corresponding components of the two vectors and sum their products. This gives the value of their dot product.
Verify the solution: Check your solution to ensure that it satisfies the given conditions in the problem. Be sure to round off the answer appropriately and include the correct units where necessary.
The angle between two vectors can be obtained from the inverse cosine of the dot product of the two vectors divided by the product of the magnitudes of the two vectors. The dot product can also be employed to find the component of a vector along a given direction by projecting it onto a unit vector in the desired direction. This technique is particularly useful for decomposing complex vector problems into simpler components. Additionally, the dot product can be used to test orthogonality between two vectors. If their dot product is zero, the vectors are orthogonal, meaning they are perpendicular to each other. Lastly, the projection of one vector onto another can be found using the dot product by multiplying the magnitude of the first vector by the cosine of the angle between the two vectors.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 71- 76.