Math Problem Statement
Let a, b and c be the vectors [1,2,0], [0, -1, 2] and [2,1,3] respectively. Find (a) a· b, (b) a· (b + c), (c) 2b· c.
Solution
Let's solve the given vector problems step by step.
Given vectors:
(a) Calculate
The dot product of two vectors and is calculated as follows:
For and :
(b) Calculate
First, find : [ \mathbf{b} + \mathbf{c} = [0, -1, 2] + [2, 1, 3] ]
Now, calculate the dot product :
(c) Calculate
First, find :
Now, calculate the dot product : [ 2\mathbf{b} \cdot \mathbf{c} = [0, -2, 4] \cdot [2, 1, 3] ]
Summary of Results
(a)
(b)
(c)
Would you like any further details or have any questions?
Here are some questions you might be interested in:
- How do you compute the cross product of two vectors?
- What is the geometric interpretation of the dot product?
- How do you find the magnitude of a vector?
- How can you determine if two vectors are orthogonal?
- What is the difference between dot product and cross product?
Tip: When computing the dot product, make sure to multiply corresponding components and sum them up correctly to avoid mistakes.
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Dot Product
Vector Addition
Formulas
Dot Product Formula: a·b = a1*b1 + a2*b2 + a3*b3
Vector Addition: b + c = [b1 + c1, b2 + c2, b3 + c3]
Theorems
Properties of Dot Product
Suitable Grade Level
Grades 11-12