Find vector c if | c | = 3 6 and c is directed along the angle bisectors of the vectors oa = 7 i – 4 j – 4 k and ob = – 2 i – j + 2 k.
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Step 1 - normalise the original vectors. So define a˙⃗ =a⃗ |a⃗ |a˙→=a→|a→| and similarly for b˙⃗ b˙→, then let c˙⃗ =a˙⃗ +b˙⃗ c˙→=a˙→+b˙→. It should be pretty simple to prove that the direction of c˙⃗ c˙→ is the same as the one of c⃗ c→ in your post.
Step 2 - Find the angle between the new proposed bisector and the original vectors. So define αα as the angle between a⃗ a→ and c⃗ c→, and then a˙⃗ ⋅c˙⃗ =|a˙⃗ ||c˙⃗ |cosα=|c˙⃗ |cosαa˙→⋅c˙→=|a˙→||c˙→|cosα=|c˙→|cosα since we set |a˙⃗ |=1|a˙→|=1 in the first step. Similarly if ββ is the angle between b⃗ b→ and c⃗ c→, then b˙⃗ ⋅c˙⃗ =|c˙⃗ |cosβb˙→⋅c˙→=|c˙→|cosβ.
But, from the way they've been defined, a˙⃗ ⋅c˙⃗ =a˙⃗ ⋅a˙⃗ +a˙⃗ ⋅b˙⃗ =|a˙⃗ |+a˙⃗ ⋅b˙⃗ =1+a˙⃗ ⋅b˙⃗ a˙→⋅c˙→=a˙→⋅a˙→+a˙→⋅b˙→=|a˙→|+a˙→⋅b˙→=1+a˙→⋅b˙→, and you can show that the other dot product has the same value. So you can conclude that cosα=cosβcosα=cosβ, and then all you have to do is show that the angles are in the same quadrant, and hence must be equal.
SHAKUNTALAM GHOSH.
Step 2 - Find the angle between the new proposed bisector and the original vectors. So define αα as the angle between a⃗ a→ and c⃗ c→, and then a˙⃗ ⋅c˙⃗ =|a˙⃗ ||c˙⃗ |cosα=|c˙⃗ |cosαa˙→⋅c˙→=|a˙→||c˙→|cosα=|c˙→|cosα since we set |a˙⃗ |=1|a˙→|=1 in the first step. Similarly if ββ is the angle between b⃗ b→ and c⃗ c→, then b˙⃗ ⋅c˙⃗ =|c˙⃗ |cosβb˙→⋅c˙→=|c˙→|cosβ.
But, from the way they've been defined, a˙⃗ ⋅c˙⃗ =a˙⃗ ⋅a˙⃗ +a˙⃗ ⋅b˙⃗ =|a˙⃗ |+a˙⃗ ⋅b˙⃗ =1+a˙⃗ ⋅b˙⃗ a˙→⋅c˙→=a˙→⋅a˙→+a˙→⋅b˙→=|a˙→|+a˙→⋅b˙→=1+a˙→⋅b˙→, and you can show that the other dot product has the same value. So you can conclude that cosα=cosβcosα=cosβ, and then all you have to do is show that the angles are in the same quadrant, and hence must be equal.
SHAKUNTALAM GHOSH.
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