Consider the following vectors: 𝐮=2 𝐢+𝐣+m 𝐤, 𝐯=3 𝐢+2 𝐣+3 𝐤, 𝐰=m 𝐢+2 𝐣+𝐤 Find a) 𝐮 ·(𝐯 ×𝐰) b) 𝐰 ·

Consider the following vectors: 𝐮=2 𝐢+𝐣+m 𝐤, 𝐯=3 𝐢+2 𝐣+3 𝐤,...

Key Concepts

Dot Product

The dot product is an operation that takes two vectors and returns a scalar. It is computed as the sum of the products of the corresponding components of the two vectors. This concept is key to projecting one vector onto another and is widely used to determine angles between vectors as well as for applications in physics such as work calculations.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in a third vector perpendicular to the plane of the original vectors. Its magnitude is given by the area of the parallelogram formed by the two vectors. The cross product is essential in problems involving torque, rotational dynamics, and determining the normal vector to a surface.

Scalar Triple Product

The scalar triple product involves three vectors and yields a scalar value by taking the dot product of one of the vectors with the cross product of the other two. This product can be used to calculate the volume of the parallelepiped defined by the three vectors, and it is sensitive to the order of the vectors, following the cyclic property. It is a crucial tool in determining coplanarity and volume in three-dimensional space.

Determinant

A determinant is a scalar value that can be computed from the elements of a square matrix. In the context of vector products, the scalar triple product can be represented as the determinant of a 3x3 matrix consisting of the components of the three vectors. Determinants play a key role in vector calculus, facilitating evaluation of volumes and solving systems of linear equations.

Transcript

00:01 Hello students here we have a question that we will we have to consider these three vectors and we have to find their triple product so we have given vector u which is 2 1 and m vector v which is 3 2 and 3 then vector w which w is equal to m 2 and 1 so we will solve it for the first problem which is you do v cross w to find you cross u dot v cross w first of all we'll find v cross w and v cross w will be equal to i j k so we see the component of v and w 3 to 3 and w components are m 2 and 1 and 2 and 1 it's equals to i into 1 into 1 minus 326 then minus j 1 z 3 minus 3 times m 3m then plus k into 3 times 2 6 minus 2 times m 2m so it's equals to minus 4 i minus m 3 j we take 3 common then plus 3 minus m 2k and now we will take 3 product of this vector which is minus 4 minus 1 minus m 3 and 2 into 3 minus m and we have vector u vector u is so u dot v cross w will be equal to vector u is given to 1 and m and m dot vector u cross v cross w which we find here is minus four minus three into one minus m and two into three minus m so students we find the dot product so you dot v cross w will be equals to 2 times 4 is minus 8 plus 1 times minus 3 k will be equals to minus 3 into 1 minus m and m times this will be close to 2m into 3 minus m so we can say that it would be equals to minus 8 minus 3 plus 3 m minus 6m plus 2m square and we can write it as m square then it will be equals to minus 3m and this is minus 11 so u .dv cross w is equals to 2m square minus 3m minus 11 now we'll find this be part of our question which is w dot u cross v w dot u cross v so first of all we will find u cross v which is equals to i j k vector u is equal to 2 1 and m and vector v is 323 3 2 3 so it will be equals to i into 3 times 1 is 3 minus 2 time m is 2m minus j into 2 3 6 minus 3m plus k into 2 2 2 2 the 4 minus 3 1's 3 so we can write it as 3 minus 2 m i minus 2 minus m into 3 j we take 3 comma from this and plus k 4 minus 3 is 1 so we can write it as 3 minus 2 m comma minus 3 into 2 minus m and 1 now we will take the dot product of w with the cross product of u and v and vector v is w is equals to m 2 and 1 so w dot u cross v w is m 2 and 1 w is m 2 and 1 and u cross v is 3 minus 2m comma let's multiply it into the bracket which gives us minus 6 and plus 3m and 1 so m times 3 minus 2m will be equals to 3m minus 2m square 2 times 2 minus 2m gives us minus 12 plus 2 times 3 is 6m and plus 1 so minus 2m square plus 3m and the 6m will be equals to 9m and minus 12 and plus 11 1 is equals to minus 11 so this is the solution for w .2 cross fee now we will solve our 3 part which is v...

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