Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to

Step 1: Express vector c in terms of a and b

Since vector $\vec{c}$ is coplanar with vectors $\vec{a}$ and $\vec{b}$, it can be expressed as a linear combination of $\vec{a}$ and $\vec{b}$. Here, $x$ and $y$ are scalar constants that we need to determine.

Step 2: Use the perpendicularity condition

We are given that $\vec{c}$ is perpendicular to $\vec{b}$. This means their dot product is zero. We substitute the expression for $\vec{c}$ and expand the dot product.

Step 3: Calculate dot products and solve for x and y

First, we calculate the dot product of $\vec{a}$ and $\vec{b}$, and the dot product of $\vec{b}$ with itself (which is the square of its magnitude). Then, we substitute these values into the equation from the previous step to find a relationship between $x$ and $y$.

Step 4: Use the given dot product condition

We are given that $\vec{a} \cdot \vec{c} = 5$. We substitute the expression for $\vec{c}$ again and expand the dot product.

Step 5: Calculate remaining dot product and solve for y

We calculate the dot product of $\vec{a}$ with itself. Then, we substitute this value and the relationship $x = -\frac{11}{2}y$ into the equation from the previous step to solve for $y$.

Step 6: Find x and vector c

Now that we have $y$, we can find $x$ using the relationship $x = -\frac{11}{2}y$. Then, we substitute the values of $x$ and $y$ back into the expression for $\vec{c}$ and simplify to find the components of $\vec{c}$.

Step 7: Calculate the magnitude of c

Finally, we calculate the magnitude of vector $\vec{c}$ using the formula $|\vec{c}| = \sqrt{c_x^2 + c_y^2 + c_z^2}$. We simplify the result.

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