The position vectors of P and Q are 5i + 4j + | vedclass

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(A) The position vectors are $\vec{OP} = 5i + 4j + ak$ and $\vec{OQ} = -i + 2j - 2k$. The vector $\vec{PQ} = \vec{OQ} - \vec{OP} = (-1 - 5)i + (2 - 4)

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$-5, 1$

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(A) The position vectors are $\vec{OP} = 5i + 4j + ak$ and $\vec{OQ} = -i + 2j - 2k$. The vector $\vec{PQ} = \vec{OQ} - \vec{OP} = (-1 - 5)i + (2 - 4)j + (-2 - a)k = -6i - 2j - (a + 2)k$. The distance between $P$ and $Q$ is given by the magnitude of $\vec{PQ}$,which is $7$. $|\vec{PQ}| = \sqrt{(-6)^2 + (-2)^2 + (-(a + 2))^2} = 7$. Squaring both sides,we get $36 + 4 + (a + 2)^2 = 49$. $40 + (a + 2)^2 = 49$. $(a + 2)^2 = 9$. $a + 2 = \pm 3$. Case $1$: $a + 2 = 3 \Rightarrow a = 1$. Case $2$: $a + 2 = -3 \Rightarrow a = -5$. Thus,the values of $a$ are $-5$ and $1$.

The position vectors of $P$ and $Q$ are $5i + 4j + ak$ and $-i + 2j - 2k$ respectively. If the distance between them is $7$,then the value of $a$ will be:

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