Question

Let \(\vec a = 2\hat i + 3\hat j + 5\hat k\), \(\vec b = \hat i - \hat j + 3\hat k\) and \(\vec c\) be a vector such that \[ \vec a \cdot \vec c = 104 \quad \text{and} \quad \vec a \times \vec c = \vec c \times \vec b. \] Then \(\vec b \cdot \vec c\) is equal to ______

Hint:

If a cross product of a sum with a vector is zero, it usually indicates parallel vectors—this greatly simplifies such problems.

Answer 1

Correct Answer: 50

Concept:
For vectors, \(\vec u \times \vec v = -(\vec v \times \vec u)\).
If \(\vec p \times \vec q = \vec 0\), then \(\vec p\) and \(\vec q\) are parallel.
A vector parallel to \(\vec v\) can be written as \(t\vec v\), where \(t\) is a scalar.
Step 1: Use the given cross product condition \[ \vec a \times \vec c = \vec c \times \vec b = -(\vec b \times \vec c) \] Hence, \[ \vec a \times \vec c + \vec b \times \vec c = \vec 0 \] \[ (\vec a + \vec b)\times \vec c = \vec 0 \] Thus, \(\vec c\) is parallel to \(\vec a+\vec b\).
Step 2: Find \(\vec a+\vec b\) \[ \vec a+\vec b = (2+1)\hat i + (3-1)\hat j + (5+3)\hat k = 3\hat i + 2\hat j + 8\hat k \] Let \[ \vec c = t(3\hat i + 2\hat j + 8\hat k) \]
Step 3: Use the dot product condition \[ \vec a \cdot \vec c = t\,[2(3)+3(2)+5(8)] \] \[ = t(6+6+40)=52t \] Given \(\vec a\cdot\vec c=104\), \[ 52t=104 \Rightarrow t=2 \] So, \[ \vec c = 6\hat i + 4\hat j + 16\hat k \]
Step 4: Find \(\vec b\cdot\vec c\) \[ \vec b\cdot\vec c = (1)(6)+(-1)(4)+(3)(16) \] \[ = 6-4+48=50 \]
Final Answer: \(\boxed{50}\)