A juggler throws a bowling pin straight up with an initial speed of 8.20 m/s. How much time elapses until the bowling pin returns to the juggler's hand?

Video transcript

Hello everyone. So in this problem we are given two vectors and is equal to five I plus four J. And n is equal to three I minus seven J. We want to find the vector product and times and in Univ actors, what is the magnitude of the vector product? So we recall the definition of the vector product and times. And simply as the determinant of the three by three matrix where the first row is I J K. Which is the unit vectors for X, Y. Z. And we can write for the next two rows, the components of the unit vectors for each of the matrix. So for em it's 54 and there is no Z or Z as a K unit vector for this expression. So we could just write zero for this role And for the same for n. 3 -7 - J and zero. Likewise for N. And looking at this matrix, we noticed that there are zeros in the column for the K or the Z. You know, vector. So we called the definition four a.m. Cross product simply as finding a vector which is perpendicular to the two vectors. We are trying to take the cross product out. So if M and n are simply in the xy planes, we know that the end times end is only going to be in the Z direction. So we only have to take the Z component of this determinant. So it would just be K times the determinant of the non zero, matrix, two by two. Matrix 5, 4, 3 -7. I recall the definition of a determinant of a two x 2 matrix is simply the difference between these cross products. So we get that five times negative seven minus four times three. And this is all in the K unit vector for the Z direction. And we get -47 K. And to find the magnitude, we know that the magnitude is equal to the square root of the X component squared, plus the Y component squared plus the z component squared. And we only have a Z component. So we know that the magnitude is the is the non zero magnitude of 47 which is simply 47. So we know that the magnitude is 47 and that the direction is in the negative Z direction. So the answer choice to this problem would be 9-2 or equal 47 and the direction is equal to negative Z. Hold themselves have a great day.