1.4.1 The Scalar Product Flashcards
The Scalar Product
- The dot product or scalar product of two vectors is defined as .
- The dot product of and is AB when the vectors have the same direction, zero when the vectors are perpendicular, and-AB when the vectors are in opposite directions.
- The dot product follows the distributive property: .
- In component form, the dot product of two vectors is
note 1
- So far we have discussed how to scale, add, and subtract vectors. What about multiplication?
- This is the definition of the scalar product or dot product.You can think of this formula as defining multiplication between two vectors and .
- You may be wondering: why is there a factor of cos θ in the formula?
- Here’s a situation that explains the definition. Suppose an object like the iguanodon is free to move in a horizontal direction. When it is pushed by a force, it moves, resulting in some displacement. The product of the push vector(technically called the force vector) and the displacement vector has to do with how much work is done pushing the object.
- But if the force does not point in the same direction as the displacement, some of the force is wasted—since the object is only free to move horizontally, only the component of force parallel to the displacement actually does anything. This component has magnitude q. It makes sense, then, that the product should equal q. - These three examples illustrate how the dot product of two vectors can be positive, zero, or negative.
- In general, if two vectors point in roughly the same direction(the angle between them is less than 90°), their dot product is positive. If they are perpendicular, the dot product is zero.If they point roughly in opposite directions (the angle between them is greater than 90°), the dot product is negative.
note 2
- What if the two vectors being considered are in component form?
- You can use the distributive property to derive a formula for the dot product of two vectors in component form. Start off by writing down the product and multiplying through to get four terms. Then use the fact that and to simplify.
- The resulting formula is simple and easy to remember.
- This example demonstrates one of the many uses of the dot product. Suppose you are given two vectors in component form. How can you find the angle between them?
- To solve this problem, combine the two formulas for the dot product. The result is a formula for the desired angle in terms of the components and magnitudes of the two vectors.
Which statement about vector A is incorrect
The value of Ay is negative.
Which part of the expression accounts for the projection of vector B in the direction of vector A
B cos θ
The angle between two vectors is 175.0°. What is the scalar product of these two vectors if their magnitudes are 4.000 cm and 3.000 cm?
−11.95
Which of the following statements about the dot product of the two vectors is not correct? Assume that one grid line is one unit.
The dot product of the two vectors is a positive value.
Which of the following answer choices related to the dot product of the two vectors is not correct?
The values Ax, Bx, Ay, and By are all greater than 0.
Which of the following statements about the dot product of the two vectors is not correct?
The value of A to two significant figures is 7.2. The value of B to two significant figures is 3.5.
A toy train is moving on a track from right to left. You are told that work is the dot product of force and displacement. In what direction would you apply a force so that you do the most negative work on the train?
You would apply a force that is in the left to right direction
Which of the following statements about scalar product is not correct?
the angle theta is defined as the angle between either vector and the horizontal axis
