Math Archive: Questions from 2024-09-19

Find the cross product a×b. a = ⟨2, 5, 0⟩, b = ⟨1, 0, 3⟩ Verify that it is orthogonal to both a and b. (a×b)⋅a = (a×b)⋅b =

Find the cross product a×b. a = 5j − 6k, b = −i + 2j + k Verify that it is orthogonal to both a and b. (a×b)⋅a = (a×b)⋅b =

Find the cross product a×b. a = ti + cos⁡(t)j + sin⁡(t)k, b = i − sin⁡(t)j + cos⁡(t)k Verify that it is orthogonal to both a and b. (a×b)⋅a = (a×b)⋅b =

Find the vector, not with determinants, but by using properties of cross products. k×(i − 9j)

State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (a) a⋅(b×c) The expression is meaningful. It is a vector. The expression is meaningful. It is a scalar. The expression is meaningless. The cross product is defined only for two vectors. The expression is meaningless. The dot product is defined only for two vectors. (b) a×(b⋅c) The expression is meaningful. It is a vector. The expression is meaningful. It is a scalar. The expression is meaningless. The cross product is defined only for two vectors. The expression is meaningless. The dot product is defined only for two vectors. (c) a×(b×c) The expression is meaningful. It is a vector. The expression is meaningful. It is a scalar. The expression is meaningless. The cross product is defined only for two vectors. The expression is meaningless. The dot product is defined only for two vectors. (d) a⋅(b⋅c) The expression is meaningful. It is a vector. The expression is meaningful. It is a scalar. The expression is meaningless. The cross product is defined only for two vectors. The expression is meaningless. The dot product is defined only for two vectors. (e) (a⋅b)×(c⋅d) The expression is meaningful. It is a vector. The expression is meaningful. It is a scalar. The expression is meaningless. The cross product is defined only for two vectors. The expression is meaningless. The dot product is defined only for two vectors. (f) (a×b)⋅(c×d) The expression is meaningful. It is a vector. The expression is meaningful. It is a scalar. The expression is meaningless. The cross product is defined only for two vectors. The expression is meaningless. The dot product is defined only for two vectors.

Find |u×v|. |v| = 7 |u×v| = Determine whether u×v is directed into the screen or out of the screen. u×v is directed into the screen. u×v is directed out of the screen.

If a = ⟨2, −1, 3⟩ and b = ⟨5, 2, 1⟩, find the following. a×b = b×a =

Find the area of the parallelogram with vertices P(2, 3, 1), Q(4, 6, 4), R(6, 12, 12), and S(4, 9, 9)

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P(−2, 1, 0), Q(4, 4, 4), R(1, 4, −1), S(3, 6, 4) cubic units

Use the scalar triple product to determine if the vectors u = i + 4j − 3k, v = 2i − j, and w = 6i + 11j − 9k are coplanar. Yes, they are coplanar. No, they are not coplanar.