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=[1,2,0]\mathbf{a} = [1, 2, 0] b=[0,1,2]\mathbf{b} = [0, -1, 2] c=[2,1,3]\mathbf{c} = [2, 1, 3]

(a) Calculate ab\mathbf{a} \cdot \mathbf{b}

The dot product of two vectors a\mathbf{a} and b\mathbf{b} is calculated as follows: ab=a1b1+a2b2+a3b3\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3

For a=[1,2,0]\mathbf{a} = [1, 2, 0] and b=[0,1,2]\mathbf{b} = [0, -1, 2]: ab=(10)+(21)+(02)\mathbf{a} \cdot \mathbf{b} = (1 \cdot 0) + (2 \cdot -1) + (0 \cdot 2) ab=02+0\mathbf{a} \cdot \mathbf{b} = 0 - 2 + 0 ab=2\mathbf{a} \cdot \mathbf{b} = -2

(b) Calculate a(b+c)\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})

First, find b+c\mathbf{b} + \mathbf{c}: [ \mathbf{b} + \mathbf{c} = [0, -1, 2] + [2, 1, 3] ] b+c=[0+2,1+1,2+3]\mathbf{b} + \mathbf{c} = [0+2, -1+1, 2+3] b+c=[2,0,5]\mathbf{b} + \mathbf{c} = [2, 0, 5]

Now, calculate the dot product a(b+c)\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}): a(b+c)=a[2,0,5]\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot [2, 0, 5] a(b+c)=(12)+(20)+(05)\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = (1 \cdot 2) + (2 \cdot 0) + (0 \cdot 5) a(b+c)=2+0+0\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 2 + 0 + 0 a(b+c)=2\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 2

(c) Calculate 2bc2\mathbf{b} \cdot \mathbf{c}

First, find 2b2\mathbf{b}: 2b=2[0,1,2]2\mathbf{b} = 2 \cdot [0, -1, 2] 2b=[02,12,22]2\mathbf{b} = [0 \cdot 2, -1 \cdot 2, 2 \cdot 2] 2b=[0,2,4]2\mathbf{b} = [0, -2, 4]

Now, calculate the dot product 2bc2\mathbf{b} \cdot \mathbf{c}: [ 2\mathbf{b} \cdot \mathbf{c} = [0, -2, 4] \cdot [2, 1, 3] ] 2bc=(02)+(21)+(43)2\mathbf{b} \cdot \mathbf{c} = (0 \cdot 2) + (-2 \cdot 1) + (4 \cdot 3) 2bc=02+122\mathbf{b} \cdot \mathbf{c} = 0 - 2 + 12 2bc=102\mathbf{b} \cdot \mathbf{c} = 10

Summary of Results

(a) ab=2\mathbf{a} \cdot \mathbf{b} = -2

(b) a(b+c)=2\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 2

(c) 2bc=102\mathbf{b} \cdot \mathbf{c} = 10

Would you like any further details or have any questions?

Here are some questions you might be interested in:

  1. How do you compute the cross product of two vectors?
  2. What is the geometric interpretation of the dot product?
  3. How do you find the magnitude of a vector?
  4. How can you determine if two vectors are orthogonal?
  5. 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