The value of r-r-r-r-r-k-r-k- is equal to

( r̂ î #160; ) ( r̂×î #160; ) +( r̂ ĵ #160; ) ( r̂×ĵ #160; ) +( r̂ k̂ #160; ) ( r̂×k̂ #160; ) =r̂( r̂×î #160; ) î( r̂×î #160; ) +r ...

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Standard XII

Physics

Vector Component

The value of ...

Question

The value of $(^r -^i) . (^r \times^i) + (^r -^j) . (^r \times^j) + (^r -^k) . (^r \times^k)$ is equal to

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Solution

$(^r -^i) (^r \times^i) + (^r -^j) (^r \times^j) + (^r -^k) (^r \times^k)$
$=^r (^r \times^i) -^i (^r \times^i) +^r (^r \times^j) -^j (^r \times^j) +^r (^r \times^k) -^k (^r \times^k)$
$= [^r^r^i] - [^i^r^i] + [^r^r^j] - [^j^r^j] + [^r^r^k] - [^k^r^k]$
$= 0$ (solar triple product $= 0$ when $2$ vectors are sence)

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The value of (^r−^i).(^r×^i)+(^r−^j).(^r×^j)+(^r−^k).(^r×^k) is equal to

(^r−^i)(^r×^i)+(^r−^j)(^r×^j)+(^r−^k)(^r×^k)=^r(^r×^i)−^i(^r×^i)+^r(^r×^j)−^j(^r×^j)+^r(^r×^k)−^k(^r×^k)=[^r^r^i]−[^i^r^i]+[^r^r^j]−[^j^r^j]+[^r^r^k]−[^k^r^k]=0 (solar triple product=0 when 2 vectors are sence)

The value of $(^r -^i) . (^r \times^i) + (^r -^j) . (^r \times^j) + (^r -^k) . (^r \times^k)$ is equal to