To find the value of
|a×(b×c)|, we need to utilize given information step by step. First, let's understand the vectors and their relationships stated in the problem.
Given:
‌||=√3‌||=5‌⋅=10The angle between
and
is
‌ is perpendicular to
×To find
|â‹…| given the angle between them and the magnitude of
, we apply the formula for dot product in terms of the angle between the vectors:
⋅=||||cos(θ)Substituting the known values, we get:
10=5||cos(‌) Since
cos(‌)=‌, we have:
‌10=5||‌‌||=4So the magnitude of
is 4 .
Given that
is perpendicular to
×, we can find the magnitude of the cross product
|×| using the formula for the cross product in terms of magnitudes and angles:
|×|=||||sin‌(θ) Substituting the known values, we get:
|×|=5⋅4⋅sin‌(‌)Since
sin‌(‌)=‌, we have:
|×|=20⋅‌=10√3 Now, to find
|×(×)|, we use the fact that the magnitude of the cross product of two vectors is given by the product of their magnitudes and the sine of the angle between them. Since
is perpendicular to
×, the angle between them is
‌, and
sin‌(‌)=1. Therefore, the magnitude of their cross product is:
‌|×(×)|=|||×|sin‌(‌)‌=√3⋅10√3⋅1=3⋅10=30