Question 20: Motion in 2D & Vectors

Question

For three vectors $$\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k}), \vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k})$$ and $$\vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k})$$, if $$\vec{A} \cdot(\vec{B} 	imes \vec{C})=0$$, then value of $$x$$ is ________.

Accepted Answer

To determine the value of $ x $, given the vectors $\vec{A}=(-x \hat{i} - 6 \hat{j} - 2 \hat{k})$, $\vec{B}=(-\hat{i} + 4 \hat{j} + 3 \hat{k})$, and $\vec{C}=(-8 \hat{i} - \hat{j} + 3 \hat{k})$, and the condition $\vec{A} \cdot (\vec{B} 	imes \vec{C}) = 0$, we proceed as follows:
First, we calculate the cross product $\vec{B} 	imes \vec{C}$:
$ \vec{B} 	imes \vec{C} = 15 \hat{i} - 21 \hat{j} + 33 \hat{k} $
Next, using the condition $\vec{A} \cdot (\vec{B} 	imes \vec{C}) = 0$, we compute the dot product:
$ \vec{A} \cdot (\vec{B} 	imes \vec{C}) = (-x \hat{i} - 6 \hat{j} - 2 \hat{k}) \cdot (15 \hat{i} - 21 \hat{j} + 33 \hat{k}) $
$ \Rightarrow (-x)(15) + (-6)(-21) + (-2)(33) = 0 $
$ \Rightarrow -15x + 126 - 66 = 0 $
Solving this equation for $ x $:
$ -15x + 60 = 0 $
$ \Rightarrow x = 4 $
Thus, the value of $ x $ is $ 4 $.

Motion in 2D & Vectors

For three vectors $\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k}), \vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k}) and \vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k}), if \vec{A} \cdot(\vec{B} \times \vec{C})=0, then value of x$ is ________.

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