Given: vector a+b+c= 0. Out of the three vectors a,b and c two are equal in magnitude. The magnitude of the third vector is √2 times that of either of the two having equal magnitude. The angles between the vectors are:

90,135,135

Let's denote the magnitude of the vectors as |a|, |b|, and |c|.

From the given information, we know that vector a + vector b + vector c = 0.

Since two of the vectors have equal magnitudes, let's assume without loss of generality that |a| = |b|.

We are also given that the magnitude of the third vector is √2 times that of either of the two vectors with equal magnitudes. So, |c| = √2 * |a|.

Now, let's find the angles between these vectors.

First, we can calculate the dot product between a and b:

a ⋅ b = |a| * |b| * cos(angle between a and b)

Since |a| = |b|, this simplifies to:

a ⋅ b = |a|^2 * cos(angle between a and b)

Similarly, the dot product between a and c is:

a ⋅ c = |a| * |c| * cos(angle between a and c)

Substituting |c| = √2 * |a|, we have:

a ⋅ c = |a| * √2 * |a| * cos(angle between a and c)

a ⋅ c = √2 * |a|^2 * cos(angle between a and c)

Since a + b + c = 0, we can rewrite this equation as:

a ⋅ c = - (a ⋅ b)

√2 * |a|^2 * cos(angle between a and c) = - |a|^2 * cos(angle between a and b)

Dividing both sides by |a|^2 gives:

√2 * cos(angle between a and c) = - cos(angle between a and b)

Similarly, we can calculate the dot product between b and c:

b ⋅ c = |b| * |c| * cos(angle between b and c)

Substituting |c| = √2 * |a|, we have:

b ⋅ c = |b| * √2 * |a| * cos(angle between b and c)

b ⋅ c = √2 * |a| * |b| * cos(angle between b and c)

Since a + b + c = 0, we can rewrite this equation as:

b ⋅ c = - (a ⋅ b)

√2 * |a| * |b| * cos(angle between b and c) = - |a|^2 * cos(angle between a and b)

Dividing both sides by |a| * |b| gives:

√2 * cos(angle between b and c) = - cos(angle between a and b)

From these two equations, we can derive the following relationship:

cos(angle between a and b) = - (√2 * cos(angle between b and c))

Now, we know that the sum of the angles between these vectors is 180 degrees (or π radians) since the vectors add up to zero.

So, we have:

(angle between a and b) + (angle between b and c) + (angle between a and c) = π

angle between a and b = π - (angle between b and c) - (angle between a and c)

Substituting the relationship between the angles that we derived earlier, we have:

angle between a and b = π - (√2 * cos(angle between b and c)) - (angle between a and c)

Therefore, the angles between the vectors are:

angle between a and b = π - (√2 * cos(angle between b and c)) - (angle between a and c)

angle between b and c = arccos(-cos(angle between a and b)/√2)

angle between a and c = angle between a and b + angle between b and c - π

Please note that the specific values of the angles will depend on the values of |a| and |b|, but this step-by-step guide can help you compute those angles.

To find the angles between the vectors a, b, and c, we need to use the dot product formula. The dot product of two vectors a and b is given by:

a · b = |a| |b| cosθ

Where |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them.

Here's how we can solve the problem step by step:

1. Given that a + b + c = 0, we can rearrange the equation to solve for one of the vectors. Let's say we solve for vector c:

c = -(a + b)

2. We know that two of the vectors, let's say a and b, have equal magnitudes. Let's call the magnitude of these vectors as X. So, |a| = |b| = X.

3. The magnitude of the third vector, c, is √2 times that of either a or b. So, |c| = √2X.

4. Now, let's calculate the dot products of these vectors to find the angles between them.

a · b = |a| |b| cos(θ1)
=> X * X * cos(θ1) = X^2 * cos(θ1)

b · c = |b| |c| cos(θ2)
=> X * √2X * cos(θ2) = X√2Xcos(θ2)

a · c = |a| |c| cos(θ3)
=> X * √2X * cos(θ3) = X√2Xcos(θ3)

5. Since a + b + c = 0, we can substitute the values of vectors a and c that we derived previously:

a · b + b · c + a · c = 0
=> X^2 * cos(θ1) + X√2Xcos(θ2) + X√2Xcos(θ3) = 0

6. From this equation, we can simplify it further by dividing each term by X^2:

cos(θ1) + √2cos(θ2) + √2cos(θ3) = 0

Now we have an equation with the angles θ1, θ2, and θ3. Unfortunately, we cannot solve for the individual angles without knowing the specific values of the cosines. However, using Trigonometry identities and equations, we can further simplify the equation.

I hope this helps in understanding how to approach the problem. Unfortunately, without specific values or additional constraints given, it is not possible to find the exact angles between the vectors a, b, and c.