If vector a and b are unit vectors, and |a+b|= root3, determine (2a-5b) dot product(b+3a)

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To determine the dot product of (2a - 5b) and (b + 3a), we first need to expand and simplify the expression.

Given:
|a + b| = √3

We can start by expanding the expression:

(2a - 5b) • (b + 3a)

= (2a • b) + (2a • 3a) + (-5b • b) + (-5b • 3a)

The dot product between two vectors is usually defined as the sum of the products of their corresponding components. So, we can rewrite the expression as:

= 2(a • b) + 2(a • a) - 5(b • b) - 15(b • a)

Now, let's simplify each term separately:

1. Simplifying a • b:
Since both a and b are unit vectors, their magnitudes are equal to 1. Therefore, a • b = |a| |b| cos θ = 1 * 1 * cos θ = cos θ, where θ is the angle between a and b.

2. Simplifying a • a:
The dot product of a vector with itself is equal to the square of its magnitude. Since a is a unit vector, its magnitude is 1. Therefore, a • a = |a|^2 = 1^2 = 1.

3. Simplifying b • b:
Similar to a • a, the dot product of a vector with itself is equal to the square of its magnitude. Since b is a unit vector, its magnitude is 1. Therefore, b • b = |b|^2 = 1^2 = 1.

4. Simplifying b • a:
The dot product of vectors is commutative, so b • a is the same as a • b. Thus, b • a = cos θ.

Substituting these values back into the expression, we have:

= 2(cos θ) + 2(1) - 5(1) - 15(cos θ)

Now, considering the given information that |a + b| = √3, we know that the magnitude of the sum of a and b is equal to √3:

|a + b| = √3
|a + b| = sqrt[(a + b) • (a + b)]
√3 = sqrt[(a + b) • (a + b)]

Expanding and simplifying this equation, we get:

3 = (a • a) + 2(a • b) + (b • b)

Since a • a = 1 and b • b = 1, this simplifies to:

3 = 1 + 2(a • b) + 1
3 = 2 + 2(a • b)

Subtracting 2 from both sides:

1 = 2(a • b)

Dividing both sides by 2:

1/2 = a • b

So, we can substitute this value back into the expression:

= 2(cos θ) + 2(1) - 5(1) - 15(cos θ)
= 2cos θ + 2 - 5 - 15cos θ

Simplifying further:

= -13cos θ - 3

Therefore, (2a - 5b) • (b + 3a) = -13cos θ - 3, where θ is the angle between vectors a and b.

To find the dot product of two vectors, we need to multiply their corresponding components and sum them up. In this case, we have the expression (2a - 5b) dot product (b + 3a). Let's break it down step by step:

Step 1: Expand the expression
(2a - 5b) dot product (b + 3a)
= (2a dot product b) + (2a dot product 3a) + (-5b dot product b) + (-5b dot product 3a)

Step 2: Simplify the dot products
The dot product of two unit vectors will be the cosine of the angle between them, which can only range from -1 to 1. Since a and b are unit vectors, their dot product will be between -1 and 1. Similarly, a dot product a is equal to 1, and b dot product b is also equal to 1.

So, we can simplify
(2a dot product b) + (2a dot product 3a) + (-5b dot product b) + (-5b dot product 3a)
= 2(a dot product b) + 2(a dot product a) - 5(b dot product b) - 5(b dot product a)

Step 3: Substitute the values we know
Since a and b are unit vectors, we can substitute their dot products as follows:
(a dot product b) = cos(theta) where theta is the angle between vectors a and b.
(a dot product a) = 1
(b dot product b) = 1

Therefore, we have:
2(a dot product b) + 2(a dot product a) - 5(b dot product b) - 5(b dot product a)
= 2(cos(theta)) + 2(1) - 5(1) - 5(cos(theta))
= 2cos(theta) + 2 - 5 - 5cos(theta)

Step 4: Simplify the expression
Combining like terms, we get:
= -3cos(theta) - 3

Now, we only need information about the angle between vectors a and b to evaluate the expression. Without that information, we cannot determine the exact value of (2a - 5b) dot product (b + 3a), and we can only express it in terms of cos(theta) as -3cos(theta) - 3.