if a and b are unit vectors, and magnitude of vectors a +b = sq. root of 3, determine (2a-5b) dot (b+3a).
(2a-5b)•(3a+b)
2a•3a - 13a•b - 5b•b
6 - 13a•b - 5
1 - 13a•b
Now |a+b| = √3, so using the law of cosines to get the angle θ between a and b, we have
1+1-2cosθ = √3
cosθ = 1-√3/2
Now, we know that since a and b are unit vectors,
a•b = cosθ = 1-√3/2
Now we have
(2a-5b)•(3a+b) = 1 - 13a•b
= 1-13(1-√3/2)
= 13√3/2 - 12
= -0.741
Well, well, well, time for some vector fun! Let's dive into it.
Given that the magnitude of the vector a + b is the square root of 3, we can express it as:
|a + b| = √3
Now let's determine the value of (2a - 5b) ⋅ (b + 3a). To do this, let's expand it:
(2a - 5b) ⋅ (b + 3a) = (2a) ⋅ b + (2a) ⋅ (3a) - (5b) ⋅ b - (5b) ⋅ (3a)
Remember that the dot product of two unit vectors is always equal to the cosine of the angle between them. Let's simplify it further:
(2a) ⋅ b = |2a| |b| cosθ₁, where θ₁ represents the angle between a and b
(2a) ⋅ (3a) = |2a| |3a| cosθ₂, where θ₂ represents the angle between 2a and 3a
(5b) ⋅ b = |5b| |b| cosθ₃, where θ₃ represents the angle between 5b and b
(5b) ⋅ (3a) = |5b| |3a| cosθ₄, where θ₄ represents the angle between 5b and 3a
Since a and b are unit vectors, the magnitudes |a| and |b| are both equal to 1. Let's simplify it further:
(2a) ⋅ b = 2 |b| cosθ₁
(2a) ⋅ (3a) = 2 |3a| cosθ₂
(5b) ⋅ b = 5 |b| cosθ₃
(5b) ⋅ (3a) = 5 |3a| cosθ₄
Now, let's consider that a and b are unit vectors, so |a| = |b| = 1. The magnitudes of 2a and 3a are therefore 2 and 3 respectively. Considering all of these simplifications, we can rewrite our expression as:
(2a - 5b) ⋅ (b + 3a) = 2 cosθ₁ + 6 cosθ₂ - 5 cosθ₃ - 15 cosθ₄
However, without any information regarding the angles θ₁, θ₂, θ₃, and θ₄, we cannot determine the numerical value of (2a - 5b) ⋅ (b + 3a). So, unfortunately, I cannot give you a specific value for that expression.
But hey, at least we had some fun with vectors along the way, right? 😄
To find the dot product of (2a - 5b) · (b + 3a), we need to expand and simplify the expression.
Let's start by rewriting the expression using the distributive property:
(2a - 5b) · (b + 3a) = (2a · b) + (2a · 3a) + (-5b · b) + (-5b · 3a)
Next, we can simplify the dot products using the properties of vectors.
Since a and b are unit vectors, their magnitudes (lengths) are both equal to 1. Therefore, we have:
a · a = 1
b · b = 1
Using these relationships, we can simplify the expression further:
(2a · b) + (2a · 3a) + (-5b · b) + (-5b · 3a)
= 2(a · b) + 6(a · a) - 5(b · b) - 15(b · a)
= 2(a · b) + 6(1) - 5(1) - 15(b · a)
= 2(a · b) + 6 - 5 - 15(b · a)
= 2(a · b) + 1 - 15(b · a)
Since we are given that the magnitude of a + b is equal to √3, we can write:
|a + b| = √3
Using the properties of vectors, we know that:
|a + b| = √((a + b) · (a + b))
Substituting the expressions for a + b and simplifying, we can rewrite the above equation as:
√3 = √((a + b) · (a + b))
= √((a · a) + 2(a · b) + (b · b))
= √(1 + 2(a · b) + 1)
= √(2 + 2(a · b))
Squaring both sides of the equation, we get:
(√3)^2 = (√(2 + 2(a · b)))^2
3 = 2 + 2(a · b)
Rearranging the equation, we can find the value of (a · b):
2(a · b) = 3 - 2
(a · b) = 1/2
Substituting this value back into the expression we want to evaluate, we have:
2(a · b) + 1 - 15(b · a)
= 2(1/2) + 1 - 15(b · a)
= 1 + 1 - 15(b · a)
= 2 - 15(b · a)
Therefore, the dot product of (2a - 5b) · (b + 3a) is equal to 2 - 15 times the dot product of b and a.
To determine the dot product of two vectors, you need to multiply their corresponding components and then sum up the results. Let's break down the given problem step by step:
1. Start with the expression (2a - 5b) dot (b + 3a).
2. Expand the expression using the distributive property: 2a dot b + 2a dot 3a - 5b dot b - 5b dot 3a.
3. Simplify the expression further: 2a dot b + 6a dot a - 5b dot b - 15b dot a.
4. Recall that the magnitude of a unit vector is always 1, so the dot product of a unit vector with itself is 1. Thus, a dot a = 1.
5. Similarly, b dot b = 1.
6. Substitute the values we derived into the expression: 2a dot b + 6(1) - 5(1) - 15b dot a.
7. Simplify the expression: 2a dot b + 6 - 5 - 15b dot a.
8. Notice that the dot product of vectors does not depend on the order, so you can rewrite a dot b as b dot a.
9. Rewrite the expression: 2b dot a + 6 - 5 - 15b dot a.
10. Combine like terms: -13b dot a + 1.
Therefore, the result of (2a - 5b) dot (b + 3a) is -13b dot a + 1.