Textbook Question
A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?
824
views
8. Vectors
Geometric Vectors
Problem 9
Textbook Question
Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.
a + b
Verified step by step guidance1
Begin by sketching vectors \( \mathbf{a} \) and \( \mathbf{b} \) on the same coordinate plane, ensuring each vector is drawn with its initial point at the origin or a common point for clarity.
Place the initial point of vector \( \mathbf{b} \) at the terminal point of vector \( \mathbf{a} \) to prepare for vector addition using the head-to-tail method.
Draw the resultant vector \( \mathbf{a} + \mathbf{b} \) starting from the initial point of \( \mathbf{a} \) to the terminal point of \( \mathbf{b} \) after it has been moved.
Alternatively, use the parallelogram rule: place both vectors \( \mathbf{a} \) and \( \mathbf{b} \) so their initial points coincide, then complete the parallelogram formed by these two vectors.
The diagonal of the parallelogram starting from the common initial point represents the vector sum \( \mathbf{a} + \mathbf{b} \). Sketch this diagonal to visualize the resultant vector.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition
Vector addition involves combining two or more vectors to find a resultant vector. This is done by placing the initial point of one vector at the terminal point of another and then drawing the vector from the start of the first to the end of the last. The resultant vector represents the combined effect of the original vectors.
Recommended video:
05:29
Adding Vectors Geometrically
Parallelogram Rule
The parallelogram rule is a geometric method to add two vectors. By placing the vectors so their tails coincide, a parallelogram is formed using the vectors as adjacent sides. The diagonal of this parallelogram from the common tail point represents the sum of the two vectors.
Recommended video:
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Vector Representation and Sketching
Accurately sketching vectors involves drawing arrows with correct direction and relative magnitude. This visual representation helps in understanding vector operations like addition. Labeling vectors clearly and maintaining scale aids in applying rules such as the parallelogram method effectively.
Recommended video:
05:05Multiplying Vectors By Scalars
Related Videos
Related Practice