a,b and c are mutually perpendicular vectors of equal magnitude. Find the angle between vector a and a+b+c
To find the angle between vector a and a+b+c, we can first find the dot product of these two vectors and then use the properties of the dot product to find the angle.
Let's start by finding the dot product of vector a and a+b+c:
a · (a+b+c) = a · a + a · b + a · c
Since a, b, and c are mutually perpendicular vectors, their dot products with each other will be zero:
a · a = |a|^2 = a^2 (magnitude of a squared)
a · b = 0
a · c = 0
So we are left with:
a · (a+b+c) = a^2 + 0 + 0
= a^2
Now, let's find the magnitudes of vector a and vector (a+b+c):
|a| = |a+b+c| (since their magnitudes are equal)
Using the definition of the dot product, we know that:
a · (a+b+c) = |a| * |a+b+c| * cosθ
where θ is the angle between vector a and vector (a+b+c).
From above, we have:
a^2 = |a| * |a+b+c| * cosθ
Dividing both sides by |a|:
a = |a+b+c| * cosθ
Dividing both sides by |a+b+c|:
a / |a+b+c| = cosθ
We can now take the inverse cosine of both sides to solve for θ:
θ = cos^(-1) (a / |a+b+c|)
So, the angle between vector a and vector (a+b+c) is given by the inverse cosine of a divided by the magnitude of (a+b+c).
This equation holds true as long as a, b, and c are mutually perpendicular vectors of equal magnitude.
To find the angle between vector a and the vector a+b+c, we can first calculate the dot product of the two vectors, and then use the dot product formula to find the angle.
Step 1: Calculate the dot product of a and (a+b+c)
The dot product of two vectors a and b is given by the formula:
a • b = |a| |b| cos(theta)
where a • b represents the dot product, |a| and |b| represent the magnitudes of vectors a and b, and theta represents the angle between them.
In this case, the magnitude of vectors a and (a+b+c) is the same since a, b, and c are mutually perpendicular vectors of equal magnitude. Therefore, we can simplify the dot product equation to:
a • (a+b+c) = |a| |a+b+c| cos(theta)
Step 2: Expand the dot product equation using vector addition
a • (a+b+c) = a • a + a • b + a • c
Step 3: Calculate the dot products
Since a, b, and c are mutually perpendicular vectors, their dot products with each other will be zero.
a • a = |a|^2 (the magnitude of a squared)
a • b = 0
a • c = 0
So, a • (a+b+c) = |a|^2
Step 4: Find the angle
Using the dot product formula:
a • (a+b+c) = |a| |a+b+c| cos(theta)
Substituting the calculations from Step 3:
|a|^2 = |a| |a+b+c| cos(theta)
Simplifying by dividing both sides by |a|:
|a| = |a+b+c| cos(theta)
Dividing both sides by |a+b+c|:
|a| / |a+b+c| = cos(theta)
Finally, using the inverse cosine (arccos) function:
theta = arccos(|a| / |a+b+c|)
Therefore, the angle between vector a and a+b+c is given by the arccosine of the ratio of the magnitudes of a and a+b+c.